The two-dimensional (2D) cine cardiovascular magnetic resonance (CMR) technique is the reference standard for assessing cardiac function. However, one challenge with 2D cine is that the acquisition time for the whole cine stack is long and requires multiple breath holds, which may not be feasible for pediatric or ill patients. Though single breath-hold multi-slice cine may address the issue, it can only acquire low-resolution images, and hence, affect the accuracy of cardiac function assessment. To address these challenges, a Ferumoxytol-enhanced, free breathing, isotropic high-resolution 3D cine technique was developed. The method produces high-contrast cine images with short acquisition times by using compressed sensing together with a manifold-based method for image denoising. This study included fifteen patients (9.1 ± 5.6 yrs.) who were referred for clinical cardiovascular magnetic resonance imaging (MRI) with Ferumoxytol contrast and were prescribed the 3D cine sequence. The data was acquired on a 1.5T scanner. Statistical analysis shows that the manifold-based denoised 3D cine can accurately measure ventricular function with no significant differences when compared to the conventional 2D breath-hold (BH) cine. The multiplanar reconstructed images of the proposed 3D cine method are visually comparable to the golden standard 2D BH cine method in terms of clarity, contrast, and anatomical precision. The proposed method eliminated the need for breath holds, reduced scan times, enabled multiplanar reconstruction within an isotropic data set, and has the potential to be used as an effective tool to access cardiovascular conditions.
Citation: Anna Andrews, Pezad Doctor, Lasya Gaur, F. Gerald Greil, Tarique Hussain, Qing Zou. Manifold-based denoising for Ferumoxytol-enhanced 3D cardiac cine MRI[J]. Mathematical Biosciences and Engineering, 2024, 21(3): 3695-3712. doi: 10.3934/mbe.2024163
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The two-dimensional (2D) cine cardiovascular magnetic resonance (CMR) technique is the reference standard for assessing cardiac function. However, one challenge with 2D cine is that the acquisition time for the whole cine stack is long and requires multiple breath holds, which may not be feasible for pediatric or ill patients. Though single breath-hold multi-slice cine may address the issue, it can only acquire low-resolution images, and hence, affect the accuracy of cardiac function assessment. To address these challenges, a Ferumoxytol-enhanced, free breathing, isotropic high-resolution 3D cine technique was developed. The method produces high-contrast cine images with short acquisition times by using compressed sensing together with a manifold-based method for image denoising. This study included fifteen patients (9.1 ± 5.6 yrs.) who were referred for clinical cardiovascular magnetic resonance imaging (MRI) with Ferumoxytol contrast and were prescribed the 3D cine sequence. The data was acquired on a 1.5T scanner. Statistical analysis shows that the manifold-based denoised 3D cine can accurately measure ventricular function with no significant differences when compared to the conventional 2D breath-hold (BH) cine. The multiplanar reconstructed images of the proposed 3D cine method are visually comparable to the golden standard 2D BH cine method in terms of clarity, contrast, and anatomical precision. The proposed method eliminated the need for breath holds, reduced scan times, enabled multiplanar reconstruction within an isotropic data set, and has the potential to be used as an effective tool to access cardiovascular conditions.
The algebraic classification (up to isomorphism) of algebras of dimension
The algebraic study of central extensions of associative and non-associative algebras has been an important topic for years (see, for example, [10,20] and references therein). Our method for classifying nilpotent commutative algebras is based on the calculation of central extensions of nilpotent algebras of smaller dimensions from the same variety (first, this method has been developed by Skjelbred and Sund for Lie algebra case in [20]) and the classifications of all complex
Throughout this paper, we use the notations and methods well written in [10], which we have adapted for the commutative case with some modifications. Further in this section we give some important definitions.
Let
Let
ϕθ(x,y)=θ(ϕ(x),ϕ(y)). |
It is easy to verify that
Let
Call the set
The following result shows that every algebra with a non-zero annihilator is a central extension of a smaller-dimensional algebra.
Lemma 1.1. Let
dim(Ann(A))=m≠0. |
Then there exists, up to isomorphism, a unique
Proof. Let
P(xy)=P((x−P(x)+P(x))(y−P(y)+P(y)))=P(P(x)P(y))=[P(x),P(y)]A′. |
Since
Definition 1.2. Let
Our task is to find all central extensions of an algebra
Let
W1=⟨[θ1],[θ2],…,[θs]⟩,W2=⟨[ϑ1],[ϑ2],…,[ϑs]⟩∈Gs(H2(A,C)), |
we easily have that if
Ts(A)={W=⟨[θ1],…,[θs]⟩∈Gs(H2(A,C)):s⋂i=1Ann(θi)∩Ann(A)=0}, |
which is stable under the action of
Now, let
E(A,V)={Aθ:θ(x,y)=s∑i=1θi(x,y)ei and ⟨[θ1],[θ2],…,[θs]⟩∈Ts(A)}. |
We also have the following result, which can be proved as in [10,Lemma 17].
Lemma 1.3. Let
Orb⟨[θ1],[θ2],…,[θs]⟩=Orb⟨[ϑ1],[ϑ2],…,[ϑs]⟩. |
This shows that there exists a one-to-one correspondence between the set of
The idea of the definition of a
((xy)a)b+((xb)a)y+x((yb)a)=((xy)b)a+((xa)b)y+x((ya)b). |
The above described method gives all commutative (
θ(x,y)=θ(y,x), |
θ((xy)a,b)+θ((xb)a,y)+θ(x,(yb)a)=θ((xy)b,a)+θ((xa)b,y)+θ(x,(ya)b). |
for all
Z2D(D,C)={θ∈Z2C(D,C):θ(x,y)=θ(y,x),θ((xy)a,b)+θ((xb)a,y)+θ(x,(yb)a)=θ((xy)b,a)+θ((xa)b,y)+θ(x,(ya)b) for all x,y,a,b∈D}. |
Observe that
Rs(D)={W∈Ts(D):W∈Gs(H2D(D,C))}, |
Us(D)={W∈Ts(D):W∉Gs(H2D(D,C))}. |
Then
Let us introduce the following notations. Let
NΞj—jth5−dimensional family ofcommutative non−CCD−algebras with parametrs Ξ.Nij—jth i−dimensional non−CCD−algebra.Ni∗j—jth i−dimensional CCD−algebra. |
Remark 1. All families of algebras from our final list do not have intersections, but inside some families of algebras there are isomorphic algebras. All isomorphisms between algebras from a certain family of algebras constucted from the representative
Thanks to [8] we have the complete classification of complex
N3∗01,N4∗01:e1e1=e2H2C=H2DN3∗02,N4∗02:e1e1=e2e1e2=e3H2C≠H2DN3∗03,N4∗03:e1e2=e3H2C=H2DN3∗04,N4∗04:e1e1=e2e2e2=e3H2C≠H2DN4∗05:e1e1=e2e1e3=e4H2C=H2DN4∗06:e1e1=e2e3e3=e4H2C=H2DN4∗07:e1e1=e4e2e3=e4H2C=H2DN4∗08:e1e1=e2e1e2=e3e2e2=e4H2C≠H2DN4∗09:e1e1=e2e2e3=e4H2C≠H2DN4∗10:e1e1=e2e1e2=e4e3e3=e4H2C≠H2DN4∗11:e1e1=e2e1e3=e4e2e2=e4H2C≠H2DN4∗12:e1e1=e2e2e2=e4e3e3=e4H2C≠H2DN4∗13(λ):e1e1=e2e1e2=e3e1e3=e4e2e2=λe4H2C≠H2DN4∗14:e1e2=e3e1e3=e4H2C≠H2DN4∗15:e1e2=e3e1e3=e4e2e2=e4H2C≠H2DN4∗16:e1e2=e3e1e3=e4e2e3=e4H2C≠H2DN4∗17:e1e2=e3e3e3=e4H2C≠H2DN4∗18:e1e1=e4e1e2=e3e3e3=e4H2C≠H2DN4∗19:e1e1=e4e1e2=e3e2e2=e4e3e3=e4H2C≠H2DN401:e1e1=e2e1e2=e3e2e3=e4N402:e1e1=e2e1e2=e3e1e3=e4e2e3=e4N403:e1e1=e2e1e2=e3e3e3=e4N404:e1e1=e2e1e2=e3e2e2=e4e3e3=e4N405:e1e1=e2e1e3=e4e2e2=e3N406:e1e1=e2e1e2=e4e1e3=e4e2e2=e3N407:e1e1=e2e2e2=e3e2e3=e4N408:e1e1=e2e1e3=e4e2e2=e3e2e3=e4N409:e1e1=e2e2e2=e3e3e3=e4N410:e1e1=e2e2e2=e3e1e2=e4e3e3=e4N411(λ):e1e1=e2e1e2=λe4e2e2=e3e2e3=e4e3e3=e4 |
Here we will collect all information about
CohomologyAutomorphismsN3∗02e1e1=e2e1e2=e3H2D(N3∗02)=⟨[Δ13],[Δ22]⟩,H2C(N3∗02)=H2D(N3∗02)⊕⟨[Δ23],[Δ33]⟩ϕ=(x00yx20z2xyx3) |
Let us use the following notations:
∇1=[Δ13],∇2=[Δ22],∇3=[Δ23],∇4=[Δ33]. |
Take
ϕT(00α10α2α3α1α3α4)ϕ=(α∗α∗∗α∗1α∗∗α∗2α∗3α∗1α∗3α∗4), |
we have
α∗1=(α1x+α3y+α4z)x3,α∗2=(α2x2+4α3xy+4α4y2)x2,α∗3=(α3x+2α4y)x4,α∗4=α4x6. |
We are interested only in
θ1=α1∇1+α2∇2+α3∇3+α4∇4 and θ2=β1∇1+β2∇2+β3∇3. |
Thus, we have
α∗1=(α1x+α3y+α4z)x3,β∗1=(β1x+β3y)x3,α∗2=(α2x2+4α3xy+4α4y2)x2,β∗2=(β2x+4β3y)x3,α∗3=(α3x+2α4y)x4,β∗3=β3x5.α∗4=α4x6. |
Consider the following cases.
x=2α24β2, y=−α3α4β2, z=α23(−2β1+β2)+2α4(α2β1−α1β2), |
we have the representatives
x=4β3α4,y=−β2α4,z=β2α3−4α1β3, |
we have the representative
x=4β1−β24β3,y=β22−4β1β216β23,z=(4β1−β2)(8β1α3β3−4β1β2α4−8α1β33+β22α4)32β33α4, |
we have the representative
x=√4α2β23−4β2α3β3+β22α44β23α4, y=−β2√α4β22−4α3β2β3+4α2β238β23√α4, z=(8β1α3β3−4β1β2α4−8α1β33+β22α4)√4α2β23−4β2α3β3+β22α416β33α4√α4, |
we have the family of representatives
Summarizing, we have the following distinct orbits:
⟨∇1,∇2+∇4⟩,⟨∇1+4∇2,−24(∇2+∇3)⟩,⟨∇1+λ∇2,∇3⟩,⟨∇1+λ∇2,∇4⟩,⟨α∇1+∇3,∇2+∇4⟩,⟨∇1+∇3,∇4⟩,⟨∇2,−3∇3⟩,⟨∇2,∇4⟩,⟨∇3,∇4⟩. |
Note that the algebras constructed from the orbits
N12:e1e1=e2e1e2=e3e1e3=e4e2e2=e5e3e3=e5N4168:e1e1=e2e1e2=e3e1e3=e4e2e2=4e4−24e5e2e3=−24e5Nλ,0170:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e2e3=e5Nλ,0184:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e3e3=e5Nα13:e1e1=e2e1e2=e3e1e3=αe4e2e2=e5e2e3=e4e3e3=e5N14:e1e1=e2e1e2=e3e1e3=e4e2e3=e4e3e3=e5N−176:e1e1=e2e1e2=e3e2e2=e4e2e3=−3e5N080:e1e1=e2e1e2=e3e2e2=e4e3e3=e5N15:e1e1=e2e1e2=e3e2e3=e4e3e3=e5 |
Here we will collect all information about
N3∗04e1e1=e2e2e2=e3H2D(N3∗04)=⟨[Δ12]⟩,H2C(N3∗04)=H2D(N3∗04)⊕⟨[Δ13],[Δ23],[Δ33]⟩ϕ=(x000x20z0x4) |
Let us use the following notations:
∇1=[Δ12],∇2=[Δ13],∇3=[Δ23],∇4=[Δ33]. |
Take
ϕT(0α1α2α10α3α2α3α4)ϕ=(α∗α∗1α∗2α∗1α∗∗α∗3α∗2α∗3α∗4), |
we have
α∗1=(α1x+α3z)x2,α∗2=(α2x+α4z)x4,α∗3=α3x5,α∗4=α4x8. |
Consider the following cases:
θ1=α1∇1+α2∇2+α3∇3+α4∇4 and θ2=β1∇1+β2∇2+β3∇3. |
Thus, we have
α∗1=(α1x+α3z)x2,β∗1=(β1x+β3z)x2,α∗2=(α2x+α4z)x4,β∗2=β2x5,α∗3=α3x6,β∗3=β3x6.α∗4=α4x8. |
Then we consider the following subcases:
x=√α3α4−1 and z=−α1√α3−1α−14, |
we have the representative
x=√β1β2 and z=(α1β2−β1α2)√β1(β1α4−β2α3)√β2, |
we have the family of representatives
2.
α∗1=(α1x+α3z)x2,β∗1=β1x3,α∗2=α2x5,β∗2=β2x5,α∗3=α3x6, |
and consider the following subcases:
3.
Summarizing, we have the following distinct orbits:
⟨∇1,∇2⟩, ⟨∇1,∇2+∇3⟩, ⟨∇1,∇3⟩, ⟨∇1,∇3+∇4⟩, ⟨∇1,∇4⟩, ⟨∇1+∇2,α∇1+∇3+∇4⟩O(α)=O(−α), ⟨∇1+∇2,∇3⟩, ⟨∇1+∇2,α∇3+∇4⟩α≠1, ⟨β∇1+∇2+∇3,α∇1+∇4⟩, ⟨α∇1+∇3,∇1+∇4⟩O(α)=O(−η3α)=O(η23α), ⟨∇1+∇3,∇4⟩, ⟨∇1+∇4,∇2⟩, ⟨∇2,∇3⟩, ⟨∇2,∇3+∇4⟩, ⟨∇2,∇4⟩, ⟨∇3,∇4⟩. |
Note that, the orbit
N076:e1e1=e2e1e2=e3e1e4=e5e2e2=e4N16:e1e1=e2e1e2=e4e1e3=e5e2e2=e3e2e3=e5N17:e1e1=e2e1e2=e4e2e2=e3e2e3=e5N18:e1e1=e2e1e2=e4e2e2=e3e2e3=e5e3e3=e5N19:e1e1=e2e1e2=e4e2e2=e3e3e3=e5Nα20:e1e1=e2e1e2=e4+αe5e1e3=e4e2e2=e3e2e3=e5e3e3=e5N21:e1e1=e2e1e2=e4e1e3=e4e2e2=e3e2e3=e5Nα≠122:e1e1=e2e1e2=e4e1e3=e4e2e2=e3e2e3=αe5e3e3=e5Nα,β23:e1e1=e2e1e2=βe4+αe5e1e3=e4e2e2=e3e2e3=e4e3e3=e5Nα24:e1e1=e2e1e2=αe4+e5e2e2=e3e2e3=e4e3e3=e5N25:e1e1=e2e1e3=e4e2e2=e3e2e3=e4e3e3=e5N26:e1e1=e2e1e2=e4e1e3=e5e2e2=e3e3e3=e4N27:e1e1=e2e1e3=e4e2e2=e3e2e3=e5N28:e1e1=e2e1e3=e4e2e2=e3e2e3=e5e3e3=e5N29:e1e1=e2e1e3=e4e2e2=e3e3e3=e5N30:e1e1=e2e2e2=e3e2e3=e4e3e3=e5 |
Here we will collect all information about
N4∗02e1e1=e2e1e2=e3H2D(N4∗02)=⟨[Δ13],[Δ22],[Δ14],[Δ24],[Δ44]⟩H2C(N4∗02)=H2D(N4∗02)⊕⟨[Δ23],[Δ33],[Δ34]⟩ϕ=(x000qx200w2xqx3re00t) |
Let us use the following notations:
∇1=[Δ13],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT(00α1α20α3α4α5α1α4α6α7α2α5α7α8)ϕ=(α∗α∗∗α∗1α∗2α∗∗α∗3α∗4α∗5α∗1α∗4α∗6α∗7α∗2α∗5α∗7α∗8), |
we have
α∗1=(α1x+α4q+α6w+α7e)x3,α∗2=(α1x+α4q+α6w+α7e)r+(α2x+α5q+α7w+α8e)t,α∗3=(α3x2+4α4xq+4α6q2)x2,α∗4=(α4x+2α6q)x4,α∗5=(α4r+α5t)x2+2(α6r+α7t)xq,α∗6=α6x6,α∗7=(α6r+α7t)x3,α∗8=α6r2+2α7rt+α8t2. |
We interested in
x=4√α2α4−α1α5,t=α24,r=−α4α5,q=−α14√α2α4−α1α5α4, |
we have the representative
x=α3−4α1α4,t=(α3−4α1)4α24(α2α4−α1α5),r=α5(α3−4α1)4α34(α1α5−α2α4),q=4α21−α1α3α24, |
we have the representative
x=α4α8,t=α34α28,q=−α1α8,r=−α24α5α28,e=α1α5−α2α4, |
we have the representative
x=α3−4α1α4,t=√(α3−4α1)5α24√α8,q=4α21−α1α3α24,r=−α5√(α3−4α1)5α34√α8,e=(4α1−α3)(α2α4−α1α5)α24α8, |
we have the representative
2.
x=2α27,q=−α5α7,e=−2α1α7,w=α25+2α1α8−2α2α7,t=−2α7,r=α8, |
we have the representative
x=1, q=−α52α7, e=−α1α7, w=α25+2α1α8−2α2α72α27, t=α3α7, r=−α3α82α27, |
we have the representative
x=√α7,t=α4,e=α3−4α14√α7,r=−α4α82α7,q=−α3√α74α4,w=4α1α4α8−4α2α4α7+α3(α5α7−α4α8)4α4√α37, |
we have the representative
x=−α32α4+α5α7−α4α82α27,q=α3(α3α27−2α4α5α7+α24α8)8α24α27,w=(α3α27−2α4α5α7+α24α8)(4α2α4α7−4α1α4α8+α3(−α5α7+α4α8))8α24α47,e=(4α1−α3)(α3α27−2α4α5α7+α24α8)8α4α37,t=(α3α27−2α4α5α7+α24α8)24α4α57,r=−α8(α3α27−2α4α5α7+α24α8)28α4α67, |
we have the representative
3.
x=1,t=α26α2α6−α1α7,q=−α42α6,r=α6α7α1α7−α2α6,e=0,w=α24−2α1α6α6, |
we have the representative
x=√α3α6−α24α26,t=√(α3α6−α24)5α36(α2α6−α1α7),q=−α4√α3α6−α242α26,r=−√(α3α6−α24)5α7α46(α2α6−α1α7),e=0, |
and
t=α26α5α6−α4α7x4, q=−α42α6x, r=α6α7α4α7−α5α6x4, e=0, w=α24−2α1α6α6x, |
we have the representatives
t=α6x3√α6α8−α27,q=−α4x2α6,r=−α7x3√α6α8−α27,e=(α1α7−α2α6)xα6α8−α27,w=(α242α26+α1α8−α2α7α27−α6α8)x, |
we have the representatives
x=α5α6−α4α7√α26(α6α8−α27),t=(α5α6−α4α7)3α26(α27−α6α8)2,q=α4(α4α7−α5α6)2α6√α26(α6α8−α27),r=α7(α4α7−α5α6)3α36(α27−α6α8)2,e=α6(α5α6−α4α7)(α4α5α6−α24α7+2α6(−α2α6+α1α7))2α36√(α6α8−α27)3,w=α6(α5α6−α4α7)(α24α8−α4α5α7+2α6(α2α7−α1α8))2α36√(α6α8−α27)3, |
we have the representative
Summarizing, we have the following distinct orbits
⟨∇2+∇3+∇4⟩,⟨∇2+α∇3+∇5+∇6⟩,⟨∇2+∇3+∇6⟩,⟨∇2+∇4⟩,⟨∇2+∇6⟩,⟨∇3+∇4+∇8⟩,⟨∇3+∇5+∇6⟩,⟨α∇3+∇5+∇6+∇8⟩,⟨∇3+∇6+∇8⟩,⟨∇3+∇7⟩,⟨∇4+∇5+∇7⟩,⟨∇4+∇7⟩,⟨∇4+∇8⟩,⟨∇5+∇6⟩,⟨∇6+∇8⟩,⟨∇7⟩, |
which gives the following new algebras:
N31:e1e1=e2e1e2=e3e1e4=e5e2e2=e5e2e3=e5Nα32:e1e1=e2e1e2=e3e1e4=e5e2e2=αe5e2e4=e5e3e3=e5N33:e1e1=e2e1e2=e3e1e4=e5e2e2=e5e3e3=e5N34:e1e1=e2e1e2=e3e1e4=e5e2e3=e5N35:e1e1=e2e1e2=e3e1e4=e5e3e3=e5N36:e1e1=e2e1e2=e3e2e2=e5e2e3=e5e4e4=e5N37:e1e1=e2e1e2=e3e2e2=e5e2e4=e5e3e3=e5Nα38:e1e1=e2e1e2=e3e2e2=αe5e2e4=e5e3e3=e5e4e4=e5N39:e1e1=e2e1e2=e3e2e2=e5e3e3=e5e4e4=e5N40:e1e1=e2e1e2=e3e2e2=e5e3e4=e5N41:e1e1=e2e1e2=e3e2e3=e5e2e4=e5e3e4=e5N42:e1e1=e2e1e2=e3e2e3=e5e3e4=e5N43:e1e1=e2e1e2=e3e2e3=e5e4e4=e5N44:e1e1=e2e1e2=e3e2e4=e5e3e3=e5N45:e1e1=e2e1e2=e3e3e3=e5e4e4=e5N46:e1e1=e2e1e2=e3e3e4=e5 |
Here we will collect all information about
N4∗04e1e1=e2e2e2=e3H2D(N4∗04)=⟨[Δ12],[Δ14],[Δ24],[Δ44]⟩,H2C(N4∗04)=H2D(N4∗04)⊕⟨[Δ13],[Δ23],[Δ33],[Δ34]⟩ϕ=(x0000x200y0x4rz00t) |
Let us use the following notations:
∇1=[Δ12],∇2=[Δ13],∇3=[Δ14],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT(0α1α2α3α10α4α5α2α4α6α7α3α5α7α8)ϕ=(α∗α∗1α∗2α∗3α∗1α∗∗α∗4α∗5α∗2α∗4α∗6α∗7α∗3α∗5α∗7α∗8), |
we have
α∗1=(α1x+α4y+α5z)x2,α∗2=(α2x+α6y+α7z)x4,α∗3=(α2x+α6y+α7z)r+(α3x+α7y+α8z)t,α∗4=α4x6,α∗5=(α4r+α5t)x2,α∗6=α6x8,α∗7=(α6r+α7t)x4,α∗8=α6r2+2α7rt+α8t2. |
We interested in
x=α2α5, t=α42α25, z=−α1α2, r=−α32α3α25, y=0, |
we have the representative
x=α3, t=√α2α28, z=−α3, r=0, y=0, |
we have the representative
x=√α1α2−1, t=4√α51α−32√α−18, z=−√α1α−12α3α−18, r=0, y=0, |
we have the representative
x=α25α2α8,t=α55α22α38,z=−α1α5α2α8,r=α45(α1α8−α3α5)α32α38,y=0, |
we have the representative
2.
3.
4.
(b)
(c)
x=5√α1α−16,t=10√α81α−36α−58,z=−α3α−185√α1α−16,y=0,r=0, |
we have the representative
x=√α4α−16, t=α24√α−36α−18, z=−α3α−18√α4α−16, y=0, r=0, |
we have the representative
Summarizing, we have the following distinct orbits:
⟨∇1+∇2+∇8⟩,⟨∇1+∇3+∇6⟩,⟨α∇1+∇3+∇4+∇6⟩O(α)=O(−α),⟨α∇1+β∇4+∇5+∇6+∇8⟩O(α,β)=O(−α,β)=O(±iα,−β),⟨α∇1+∇4+∇6+∇8⟩O(α)=O(−α),⟨∇1+∇4+∇7⟩,⟨α∇1+∇4+∇5+∇7⟩O(α)=O(−α),⟨∇1+∇5+∇7⟩,⟨∇1+∇6+∇8⟩,⟨∇1+∇7⟩,⟨∇2+∇3+∇4⟩,⟨∇2+∇4+∇8⟩,⟨∇2+∇5⟩,⟨∇2+∇5+∇8⟩,⟨∇2+∇8⟩,⟨∇3+∇4⟩,⟨α∇3+∇4+∇5+∇6⟩O(α)=O(−α),⟨∇3+∇5+∇6⟩,⟨∇3+∇6⟩,⟨∇4+∇7⟩,⟨∇4+∇8⟩,⟨∇5+∇6⟩,⟨∇5+∇7⟩,⟨∇6+∇8⟩,⟨∇7⟩. |
Hence, we have the following new algebras:
N47:e1e1=e2e1e2=e5e1e3=e5e2e2=e3e4e4=e5N48:e1e1=e2e1e2=e5e1e4=e5e2e2=e3e3e3=e5Nα49:e1e1=e2e1e2=αe5e1e4=e5e2e2=e3e2e3=e5e3e3=e5Nα,β50:e1e1=e2e1e2=αe5e2e2=e3e2e3=βe5e2e4=e5e3e3=e5e4e4=e5Nα51:e1e1=e2e1e2=αe5e2e2=e3e2e3=e5e3e3=e5e4e4=e5N52:e1e1=e2e1e2=e5e2e2=e3e2e3=e5e3e4=e5Nα53:e1e1=e2e1e2=αe5e2e2=e3e2e3=e5e2e4=e5e3e4=e5N54:e1e1=e2e1e2=e5e2e2=e3e2e4=e5e3e4=e5N55:e1e1=e2e1e2=e5e2e2=e3e3e3=e5e4e4=e5N56:e1e1=e2e1e2=e5e2e2=e3e3e4=e5N57:e1e1=e2e1e3=e5e1e4=e5e2e2=e3e2e3=e5N58:e1e1=e2e1e3=e5e2e2=e3e2e3=e5e4e4=e5N59:e1e1=e2e1e3=e5e2e2=e3e2e4=e5N60:e1e1=e2e1e3=e5e2e2=e3e2e4=e5e4e4=e5N61:e1e1=e2e1e3=e5e2e2=e3e4e4=e5N62:e1e1=e2e1e4=e5e2e2=e3e2e3=e5Nα63:e1e1=e2e1e4=αe5e2e2=e3e2e3=e5e2e4=e5e3e3=e5N64:e1e1=e2e1e4=e5e2e2=e3e2e4=e5e3e3=e5N65:e1e1=e2e1e4=e5e2e2=e3e3e3=e5N66:e1e1=e2e2e2=e3e2e3=e5e3e4=e5N67:e1e1=e2e2e2=e3e2e3=e5e4e4=e5N68:e1e1=e2e2e2=e3e2e4=e5e3e3=e5N69:e1e1=e2e2e2=e3e2e4=e5e3e4=e5N70:e1e1=e2e2e2=e3e3e3=e5e4e4=e5N71:e1e1=e2e2e2=e3e3e4=e5 |
Here we will collect all information about
N4∗08e1e1=e2e1e2=e3e2e2=e4H2D(N4∗08)=⟨[Δ13],[Δ14]+3[Δ23]⟩H2C(N4∗08)=H2D(N4∗08)⊕⟨[Δ14],[Δ24],[Δ33],[Δ34],[Δ44]⟩ϕ=(x000yx200z2xyx30ty2x2yx4) |
Let us use the following notations:
∇1=[Δ13],∇2=[Δ14]+3[Δ23],∇3=[Δ14],∇4=[Δ24],∇5=[Δ33],∇6=[Δ34],∇7=[Δ44]. |
Take
ϕT(00α1α2+α3003α2α4α13α2α5α6α2+α3α4α6α7)ϕ=(α∗α∗∗α∗1α∗2+α∗3α∗∗α∗∗∗3α∗2α∗4α∗13α∗2α∗5α∗6α∗2+α∗3α∗4α∗6α∗7), |
we have
α∗1=(α1x+3α2y+α5z+α6t)x3+((α2+α3)x+α4y+α6z+α7t)x2y,α∗2=13(3α2x3+(α4+2α5)x2y+3α6xy2+α7y3)x2,α∗3=((α2+α3)x+α4y+α6z+α7t)x4−13(3α2x3+(α4+2α5)x2y+3α6xy2+α7y3)x2,α∗4=(α4x2+2α6xy+α7y2)x4,α∗5=(α5x2+2α6xy+α7y2)x4,α∗6=(α6x+α7y)x6,α∗7=α7x8. |
We are interested in
⟨−14∇2+∇3⟩ and ⟨∇1−14∇2+∇3⟩ |
depending on
2.
3.
x=α2+α3α5, y=3α2α3+3α232α25, z=−(α2+α3)(2α1α5+12α2α3+3α23)4α35, |
we have the representative
x=2(α4−α5),y=3α3,z=0,t=0, |
we have the representative
x=2(α2α5−α2α4+α3α5)+α3α42(α25−α4α5),y=3α3(2(α2α5−α2α4+α3α5)+α3α4)2α5(α5−α4)2,z=−(2α2(α4−α5)−α3(α4+2α5))(4α1(α4−α5)2−24α2α3(α4−α5)+3α23(α4+2α5))8(α4−α5)3α25,t=0, |
we have the family of representatives
4. if
z=y2−α3α6+2y(α5−α4)3α6 and t=−x2α1+xy(4α2+α3)+xzα5+y(yα4+zα6)α6), |
we have
5. if
Summarizing all cases we have the following distinct orbits
⟨∇1−14∇2+∇3⟩,⟨α∇1+∇2+β∇4+∇7⟩O(α,β)=O(−η3α,η23β)=O(η23α,−η3β),⟨∇1+α∇3+∇4⟩O(α)=O(−α),⟨∇1+α∇4+∇7⟩O(α)=O(−α),⟨α∇2+∇3⟩α≠0,−1,⟨∇2+α∇4+∇5⟩,⟨α∇2+β∇4+∇5+∇7⟩O(α,β)=O(−α,β),⟨∇3+∇4+∇5⟩,⟨α∇4+∇5⟩α≠0,1,⟨∇4+α∇5+∇6⟩,⟨∇6⟩, |
which gives the following new algebras:
N72:e1e1=e2e1e2=e3e1e3=e5e1e4=34e5e2e2=e4e2e3=−34e5Nα,β73:e1e1=e2e1e2=e3e1e3=αe5e1e4=e5e2e2=e4e2e3=3e5e2e4=βe5e4e4=e5Nα74:e1e1=e2e1e2=e3e1e3=e5e1e4=αe5e2e2=e4e2e4=e5Nα75:e1e1=e2e1e2=e3e1e3=e5e2e2=e4e2e4=αe5e4e4=e5Nα≠0,−176:e1e1=e2e1e2=e3e1e4=(1+α)e5e2e2=e4e2e3=3αe5Nα77:e1e1=e2e1e2=e3e1e4=e5e2e2=e4e2e3=3e5e2e4=αe5e3e3=e5Nα,β78:e1e1=e2e1e2=e3e1e4=αe5e2e2=e4e2e3=3αe5e2e4=βe5e3e3=e5e4e4=e5N79:e1e1=e2e1e2=e3e1e4=e5e2e2=e4e2e4=e5e3e3=e5Nα≠0,180:e1e1=e2e1e2=e3e2e2=e4e2e4=αe5e3e3=e5Nα81:e1e1=e2e1e2=e3e2e2=e4e2e4=e5e3e3=αe5e3e4=e5N82:e1e1=e2e1e2=e3e2e2=e4e3e4=e5 |
Here we will collect all information about
N4∗09e1e1=e2e2e3=e4H2D(N4∗09)=⟨[Δ12],[Δ13],[Δ22],[Δ33]⟩H2C(N4∗09)=H2D(N4∗09)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩ϕ=(x0000x20000r0t0sx2r) |
Let us use the following notations:
∇1=[Δ12],∇2=[Δ13],∇3=[Δ14],∇4=[Δ22],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT(0α1α2α3α1α40α5α20α6α7α3α5α7α8)ϕ=(α∗α∗1α∗2α∗3α∗1α∗4α∗∗α∗5α∗1α∗∗α∗6α∗7α∗3α∗5α∗7α∗8), |
we have
α∗1=(α1x+α5t)x2,α∗2=(α2x+α7t)r+(α3x+α8t)s,α∗3=(α3x+α8t)x2r,α∗4=α4x4,α∗5=α5x4r,α∗6=(α6r+α7s)r+(α7r+α8s)s,α∗7=(α7r+α8s)x2r,α∗8=α8r2x4. |
We are interested in
x=α1α−14, r=α1α−13, s=−α1α2α−23, t=0, |
we have the family of representatives
2.
x=4√α4α6α−25, r=α4α5−1, t=−α14√α4α6α−65,s=0, |
we have the representative
r=1,x=3√α2α5−1,t=−α13√α2α−45,s=0, |
we have the representative
x=3√α2α5−1, r=α−163√α42α−15, t=−α13√α2α−45,s=0, |
we have the representative
x=3√α2α−15,r=α4α5−1,t=−α13√α2α−45,s=0, |
we have the family of representatives
x=α3α5−1,r=α43α−35α−16,s=−α2α33α−35α−16,t=−α1α3α−25, |
we have the representative
3.
4.
Summarizing, we have the following distinct orbits:
⟨∇1+∇2+α∇4+β∇5+γ∇6+∇8⟩O(α,β,γ)=O(−η5α,−η35β,−η5γ)=O(η25α,−η5β,η25γ)=O(−η35α,η45β,−η35γ)=O(η45α,η25β,η45γ),⟨∇1+∇3⟩,⟨∇1+∇3+α∇4+β∇5+∇7⟩,⟨∇1+∇3+∇4+α∇6⟩,⟨∇1+∇3+∇6⟩,⟨∇1+∇4+α∇5+β∇6+∇8⟩O(α,β)=O(−α,β),⟨∇1+∇4+α∇5+∇7⟩O(α,β)=O(−α,β),⟨∇1+∇5+α∇6+∇8⟩,⟨∇1+∇5+∇7⟩,⟨∇1+∇6+∇8⟩,⟨∇1+∇7⟩,⟨∇1+∇8⟩,⟨∇2+∇4+∇5+α∇6⟩O(α)=O(η3α)=O(η23α),⟨∇2+∇4+α∇5+β∇6+∇8⟩,⟨∇2+∇5⟩,⟨∇2+∇5+∇6⟩,⟨∇2+∇5+α∇6+∇8⟩O(α,β)=O(−α,β)=O(α,η23β)=O(−α,η23β)=O(−α,−η3β)=O(α,−η3β),⟨∇2+∇6+∇8⟩,⟨∇2+∇8⟩,⟨∇3⟩,⟨∇3+∇4⟩,⟨∇3+∇4+∇5+α∇6⟩,⟨∇3+α∇4+∇5+∇7⟩,⟨∇3+∇4+∇6⟩,⟨∇3+∇5⟩,⟨∇3+∇5+∇6⟩,⟨∇3+∇6⟩,⟨∇4+∇5⟩,⟨∇4+∇5+∇6⟩,⟨∇4+α∇5+∇6+∇8⟩O(α)=O(−α),⟨∇4+∇5+∇7⟩,⟨∇4+∇5+∇8⟩,⟨∇4+∇7⟩,⟨∇4+∇8⟩,⟨∇5⟩,⟨∇5+∇6⟩,⟨∇5+∇6+∇8⟩,⟨∇5+∇7⟩,⟨∇5+∇8⟩,⟨∇6+∇8⟩,⟨∇7⟩,⟨∇8⟩, |
which gives the following new algebras:
Nα,β,γ83:e1e1=e2e1e2=e5e1e3=e5e2e2=αe5e2e3=e4e2e4=βe5e3e3=γe5e4e4=e5N84:e1e1=e2e1e2=e5e1e4=e5e2e3=e4Nα,β85:e1e1=e2e1e2=e5e1e4=e5e2e2=αe5e2e3=e4e2e4=βe5e3e4=e5Nα86:e1e1=e2e1e2=e5e1e4=e5e2e2=e5e2e3=e4e3e3=αe5N87:e1e1=e2e1e2=e5e1e4=e5e2e3=e4e3e3=e5Nα,β88:e1e1=e2e1e2=e5e2e2=e5e2e3=e4e2e4=αe5e3e3=βe5e4e4=e5Nα89:e1e1=e2e1e2=e5e2e2=e5e2e3=e4e2e4=αe5e3e4=e5Nα90:e1e1=e2e1e2=e5e2e3=e4e2e4=e5e3e3=αe5e4e4=e5N91:e1e1=e2e1e2=e5e2e3=e4e2e4=e5e3e4=e5N92:e1e1=e2e1e2=e5e2e3=e4e3e3=e5e4e4=e5N93:e1e1=e2e1e2=e5e2e3=e4e3e4=e5N94:e1e1=e2e1e2=e5e2e3=e4e4e4=e5Nα95:e1e1=e2e1e3=e5e2e2=e5e2e3=e4e2e4=e5e3e3=αe5Nα,β96:e1e1=e2e1e3=e5e2e2=e5e2e3=e4e2e4=αe5e3e3=βe5e4e4=e5N97:e1e1=e2e1e3=e5e2e3=e4e2e4=e5N98:e1e1=e2e1e3=e5e2e3=e4e2e4=e5e3e3=e5Nα99:e1e1=e2e1e3=e5e2e3=e4e2e4=e5e3e3=αe5e4e4=e5N100:e1e1=e2e1e3=e5e2e3=e4e3e3=e5e4e4=e5N101:e1e1=e2e1e3=e5e2e3=e4e4e4=e5N102:e1e1=e2e1e4=e5e2e3=e4N103:e1e1=e2e1e4=e5e2e2=e5e2e3=e4Nα104:e1e1=e2e1e4=e5e2e2=e5e2e3=e4e2e4=e5e3e3=αe5Nα105:e1e1=e2e1e4=e5e2e2=αe5e2e3=e4e2e4=e5e3e4=e5N106:e1e1=e2e1e4=e5e2e2=e5e2e3=e4e3e3=e5N107:e1e1=e2e1e4=e5e2e3=e4e2e4=e5N108:e1e1=e2e1e4=e5e2e3=e4e2e4=e5e3e3=e5N109:e1e1=e2e1e4=e5e2e3=e4e3e3=e5N110:e1e1=e2e2e2=e5e2e3=e4e2e4=e5N111:e1e1=e2e2e2=e5e2e3=e4e2e4=e5e3e3=e5Nα112:e1e1=e2e2e2=e5e2e3=e4e2e4=αe5e3e3=e5e4e4=e5N113:e1e1=e2e2e2=e5e2e3=e4e2e4=e5e3e4=e5N114:e1e1=e2e2e2=e5e2e3=e4e2e4=e5e4e4=e5N115:e1e1=e2e2e2=e5e2e3=e4e3e4=e5N116:e1e1=e2e2e2=e5e2e3=e4e4e4=e5N117:e1e1=e2e2e3=e4e2e4=e5N118:e1e1=e2e2e3=e4e2e4=e5e3e3=e5N119:e1e1=e2e2e3=e4e2e4=e5e3e3=e5e4e4=e5N120:e1e1=e2e2e3=e4e2e4=e5e3e4=e5N121:e1e1=e2e2e3=e4e2e4=e5e4e4=e5N122:e1e1=e2e2e3=e4e3e3=e5e4e4=e5N123:e1e1=e2e2e3=e4e3e4=e5N124:e1e1=e2e2e3=e4e4e4=e5 |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{10} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_4 \\ e_3e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{10}) = \\ \Big\langle \begin{array}{c} [\Delta_{13}],[\Delta_{14}],[\Delta_{22}], \\ [\Delta_{23}],[\Delta_{33}] \end{array} \Big\rangle\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{10}) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{10})\oplus\\ {\langle [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle} \end{array} & \begin{array}{l} { \phi = \begin{pmatrix} x&0&0&0\\ y&x^2&-\frac{zr}{x}&0\\ z&0&r&0\\ t&z^2+2xy&s&x^3 \end{pmatrix}} , \\ {r^2 = x^3} \end{array}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ \alpha_1&\alpha_4&\alpha_6&\alpha_7\\ \alpha_2&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1&\alpha_2^*\\ \alpha^{**}&\alpha_3^{*}&\alpha^*_4&\alpha^*_5\\ \alpha^{*}_1&\alpha^*_4&\alpha^*_6+\alpha^{**}&\alpha^*_7\\ \alpha^*_2&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix}, |
we have
\begin{array}{lcl} \alpha_1^* & = & -(\alpha_3y+\alpha_4z+\alpha_5t)\frac{zr}{x}+(\alpha_1x+\alpha_4y+\alpha_6z+\alpha_7t)r+ \\ && (\alpha_2x+\alpha_5y+\alpha_7z+\alpha_8t)s, \\ \alpha_2^* & = & (\alpha_2x+\alpha_5y+\alpha_7z+\alpha_8t)x^3, \\ \alpha_3^* & = & \alpha_3x^4+2\alpha_5x^2(z^2+2xy)+\alpha_8(z^2+2xy)^2, \\ \alpha_4^* & = & -(\alpha_3x^2+\alpha_5(z^2+2xy))\frac{zr}{x}+(\alpha_4x^2+\alpha_7(z^2+2xy))r+ \\&& (\alpha_5x^2+\alpha_8(z^2+2xy))s, \\ \alpha_5^* & = & (\alpha_5x^2+\alpha_8(z^2+2xy))x^3, \\ \alpha_6^* & = & -(\alpha_4r-\alpha_3\frac{zr}{x}+\alpha_5s)\frac{zr}{x}+(\alpha_6r-\alpha_4\frac{zr}{x}+\alpha_7s)r+ \\ && (\alpha_7r-\alpha_5\frac{zr}{x}+\alpha_8s)s-(\alpha_3y+\alpha_4z+\alpha_5t)x^2- \\ && (\alpha_2x+\alpha_5y+\alpha_7z+\alpha_8t)(z^2+2xy), \\ \alpha_7^* & = & (\alpha_7r-\alpha_5\frac{zr}{x}+\alpha_8s)x^3, \\ \alpha_8^* & = & \alpha_8x^6. \end{array} |
We are interested in
\begin{array}{c} y = -\frac{ \alpha_2^2+\alpha_2\alpha_3+\alpha_4\alpha_7}{2\alpha_7^2}x, z = -\frac{\alpha_2}{\alpha_7}x, \\s = -\frac{3\alpha^2_2\alpha_3+\alpha_2(\alpha_3^2+6\alpha_4 \alpha_7)+\alpha_7(\alpha_3\alpha_4+2\alpha_6\alpha_7)}{4\alpha_7^3}\sqrt{x^3},\\ t = \frac{\alpha_7^2(\alpha_4^2-2\alpha_1\alpha_7)+\alpha_2^3\alpha_3+\alpha_2^2(\alpha_3^2+3\alpha_4\alpha_7)+2\alpha_2\alpha_7 (\alpha_3\alpha_4+\alpha_6\alpha_7)}{2\alpha_7^4}x, \end{array} |
we have
\begin{array}{c} y = -\frac{\alpha_2\alpha_5+\alpha_7^2}{\alpha_5^2}x, z = \frac{\alpha_7}{\alpha_5}x,\\s = \frac{\alpha_3\alpha_7-\alpha_4\alpha_5}{\alpha^2_5}\sqrt{x^{3}},\\ t = \frac{\alpha_2\alpha_3\alpha_5+\alpha_5^2\alpha_6+3\alpha_4\alpha_5\alpha_7-2\alpha_3\alpha^2_7}{\alpha^3_5}x, \end{array} |
we have
3.
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+\alpha \nabla_3+\nabla_5 \rangle ^{O(\alpha) = O(\eta^4_5\alpha) = O(-\eta^3_5\alpha) = O(\eta^2_5\alpha) = O(-\eta_5\alpha)}, \, \\ \langle \alpha\nabla_1+ \nabla_3+\beta \nabla_6 + \nabla_8 \rangle^{{ \begin{array}{l} O(\alpha,\beta) = O(-\alpha,\beta) = O(\eta_3\alpha,\eta_3^2\beta) = \\ O(-\eta_3\alpha,\eta_3^2\beta) = O(-\eta^2_3\alpha,-\eta_3\beta) = O(\eta_3^2\alpha,-\eta_3\beta) \end{array}}}, \\ \langle \alpha\nabla_1+ \nabla_4+\nabla_8\rangle^ {{ \begin{array}{l} O(\alpha) = O(-\alpha) = O(\eta_5^4\alpha) = O(-\eta_5^4\alpha) = O(\eta_5^3\alpha) = \\O(-\eta_5^3\alpha) = O(\eta_5^2\alpha) = O(-\eta_5^2\alpha) = O(\eta_5\alpha) = O(-\eta_5\alpha) \end{array}}}, \, \\ \langle \nabla_1+ \nabla_8 \rangle, \langle \nabla_3+ \nabla_5 \rangle, \langle \nabla_3+\nabla_7 \rangle, \langle \nabla_5 \rangle, \langle \nabla_6+\nabla_8 \rangle, \langle \nabla_7\rangle, \langle \nabla_8\rangle, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}^{\alpha}_{125} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}^{\alpha,\beta}_{126} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 \\ && e_2e_2 = e_5 & e_3e_3 = e_4 + \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}^{\alpha}_{127} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 \\ && e_2e_3 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{128} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{129} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{130} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{131} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{132} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_3e_3 = e_4+e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{133} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{134} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{11} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_3 = e_4 \\ e_2e_2 = e_4 \end{array} & \begin{array}{lcl} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{11}) & = & \\ {\langle [\Delta_{12}],[\Delta_{22}],[\Delta_{23}],[\Delta_{33}]\rangle }\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{11})& = &\mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{11})\oplus\\ {\langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}]\rangle} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ z&0&x^3&0\\ t&2xz&s&x^4 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&\alpha_1&0&\alpha_2\\ \alpha_1&\alpha_3&\alpha_4&\alpha_5\\ 0&\alpha_4&\alpha_6&\alpha_7\\ \alpha_2&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha_1^{*}&\alpha^{**}&\alpha_2^*\\ \alpha_1^{*}&\alpha_3^{*}+\alpha^{**}&\alpha^*_4&\alpha^*_5\\ \alpha^{**}&\alpha^*_4&\alpha^*_6&\alpha^*_7\\ \alpha^*_2&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix}, |
we have
\begin{array}{lcl} \alpha_1^* & = & (\alpha_1x+\alpha_4z+\alpha_5t)x^2+2(\alpha_2x+\alpha_7z+\alpha_8t)xz, \\ \alpha_2^* & = & (\alpha_2x+\alpha_7z+\alpha_8t)x^4, \\ \alpha_3^* & = & (\alpha_3x^2+4\alpha_5xz+4\alpha_8z^2)x^{2}-(\alpha_6z+\alpha_7t)x^3-(\alpha_2x+\alpha_7z+\alpha_8t)s, \\ \alpha_4^* & = & (\alpha_4x+2\alpha_7z)x^4+(\alpha_5x+2\alpha_8z)xs, \\ \alpha_5^* & = & (\alpha_5x+2\alpha_8z)x^5, \\ \alpha_6^* & = & \alpha_6x^6+2\alpha_7x^3s+\alpha_8s^2, \\ \alpha_7^* & = & (\alpha_7x^3+\alpha_8s)x^4, \\ \alpha_8^* & = & \alpha_8x^8. \end{array} |
We are interested in
x = \sqrt{\frac{\alpha_1}{\alpha_2}}, z = 0, s = \frac{\alpha_1\alpha_3\sqrt{\alpha_1}}{\alpha_2^2\sqrt{\alpha_2}}, t = 0, |
we have the family of representatives
x = \alpha_4+2\alpha_2, z = -\alpha_1, s = \frac{\alpha_3(\alpha_4+2\alpha_2)^3}{\alpha_2}, t = 0, |
we have the the family of representatives
x = \frac{\alpha_2}{\alpha_6}, z = -\frac{\alpha_1\alpha_2}{\alpha_6(\alpha_4+2\alpha_2)}, s = \frac{\alpha_1\alpha^2_2\alpha_6+2\alpha_2^3\alpha_3+\alpha_2^2\alpha_3\alpha_4}{\alpha_6^3(\alpha_4+2\alpha_2)}, t = 0, |
we have the family of representatives
2.
x = 4\alpha_5, z = -\alpha_3, s = -64\alpha_4\alpha_5^2, t = \frac{\alpha_3\alpha_4-4\alpha_1\alpha_5}{\alpha_5} |
we have the representative
\begin{array}{c} x = \frac{\alpha_2}{\alpha_5}, z = -\frac{\alpha_2^2\alpha_4+\alpha_2\alpha_3\alpha_5}{4\alpha_5^3}, s = -\frac{\alpha_2^3\alpha_4}{\alpha_5^4}, \\ t = \frac{(\alpha_2\alpha_4+2\alpha^2_2)(\alpha_2\alpha_4+\alpha_3\alpha_5)-4\alpha_1\alpha_2\alpha_5^2}{4\alpha_5^4},\end{array} |
we have the representative
x = \frac{\alpha_2}{\alpha_5}, z = 0, s = -\frac{\alpha_2^3\alpha_4}{\alpha_5^4}, t = -\frac{\alpha_1\alpha_2}{\alpha_5^2}, |
we have the representative
x = \frac{\sqrt{\alpha_2\alpha_4+\alpha_3\alpha_5}}{\alpha_5}, z = 0, s = -\frac{\alpha_4\sqrt{(\alpha_2\alpha_4+\alpha_3\alpha_5)^3}}{\alpha^4_5}, t = -\frac{\alpha_1\sqrt{\alpha_2\alpha_4+\alpha_3\alpha_5}}{\alpha^2_5}, |
we have the family of representatives
x = \alpha_6-4\alpha_5, z = \alpha_3, s = \frac{\alpha_4(4\alpha_5-\alpha_6)^3}{\alpha_5}, t = \frac{4\alpha_1\alpha_5-\alpha_1\alpha_6-\alpha_3\alpha_4}{\alpha_5}, |
we have the family of representatives
\begin{array}{c} x = \frac{\alpha_2}{\alpha_5}, z = \frac{\alpha_2 (\alpha_2 \alpha_4+\alpha_3 \alpha_5)}{\alpha_5^2\alpha_6-4\alpha_5^2}, s = -\frac{\alpha_4\alpha_2^3}{\alpha^4_5}, \\ t = \frac{\alpha_2 (2 \alpha_2^2 \alpha_4+\alpha_3 \alpha_4 \alpha_5+\alpha_2 (\alpha_4^2+2 \alpha_3 \alpha_5)-\alpha_1 \alpha_5 (4 \alpha_5-\alpha_6))}{\alpha_5^3 (4 \alpha_5-\alpha_6)}, \end{array} |
we have the family of representatives
3.
4.
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+ \nabla_2 - 2 \nabla_4 + \alpha\nabla_6\rangle^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_1+ \alpha\nabla_3 +\beta \nabla_4+\gamma \nabla_6 + \nabla_8 \rangle^{ { \begin{array}{l} O(\alpha,\beta,\gamma) = O(-\eta_5\alpha,\eta_5^2\beta,-\eta_5^3\gamma) = O(\eta_5^2\alpha,\eta_5^4\beta,-\eta_5\gamma) = \\ O(-\eta_5^3\alpha,-\eta_5\beta,\eta_5^4\gamma) = O(\eta_5^4\alpha,-\eta_5^3\beta,\eta_5^2\gamma)\end{array}}}, \\ \langle \nabla_1+\alpha\nabla_4+\beta\nabla_5+\nabla_7 \rangle ^{O(\alpha,\beta) = O(\alpha,-\beta) = O(-\alpha,-i\beta) = O(-\alpha,i\beta)}, \\ \langle \alpha\nabla_2+ \nabla_3+\nabla_5 + 4\nabla_6 \rangle^{O(\alpha) = O(-\alpha)}, \langle \nabla_2+ \alpha\nabla_4 \rangle, \langle \nabla_2+ \alpha\nabla_4 + \nabla_6 \rangle, \\ \langle \nabla_2+ \nabla_5 + \alpha\nabla_6 \rangle, \langle \nabla_3+ \alpha\nabla_4+\beta\nabla_6 + \nabla_8 \rangle, \langle \nabla_4+ \alpha\nabla_5+\nabla_7 \rangle, \\ \langle \nabla_4+ \alpha\nabla_6+\nabla_8 \rangle, \langle \nabla_5+ \alpha\nabla_6 \rangle, \langle \nabla_5+ \nabla_7 \rangle, \langle \nabla_6+ \nabla_8 \rangle, \langle \nabla_7 \rangle, \langle \nabla_8 \rangle,\end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{135}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_3 = -2e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{136}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_4+\alpha e_5 \\ & & e_2e_3 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{137}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = \alpha e_5 & e_2e_4 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{138}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4+e_5 & e_2e_4 = e_5 & e_3e_3 = 4e_5 \\ {\mathbf{N}}_{139}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{140}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{141}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\& & e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{142}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4+e_5 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{143}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{144}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{145}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{146} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{147} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{148} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{149} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{12} & \begin{array}{l} e_1e_1 = e_2 \\ e_2e_2 = e_4 \\ e_3e_3 = e_4 \end{array} & \begin{array}{lcl} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{12})& = &\\ {\langle [\Delta_{12}],[\Delta_{13}],[\Delta_{23}],[\Delta_{33}]\rangle}\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{12})& = &\mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{12})\oplus\\ {\langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle } \end{array} & \phi_{\pm} = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&0& \pm x^2&0\\ t&0&s&x^4 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array} |
Take
\phi_{\pm}^T\begin{pmatrix} 0&\alpha_1&\alpha_2&\alpha_3\\ \alpha_1&0&\alpha_4&\alpha_5\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7\\ \alpha_3&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi_{\pm} = \begin{pmatrix} \alpha^*&\alpha_1^{*}&\alpha^{*}_2&\alpha_3^*\\ \alpha_1^{*}&0&\alpha^*_4&\alpha^*_5\\ \alpha^{*}_2&\alpha^*_4&\alpha^*_6&\alpha^*_7\\ \alpha^*_3&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix}, |
we have
\begin{array}{ll} \alpha_1^* = (\alpha_1x+\alpha_5t)x^2, & \alpha_2^* = (\alpha_3x+\alpha_8t)s\pm(\alpha_2x+\alpha_7t)x^2, \\ \alpha_3^* = (\alpha_3x+\alpha_8t)x^4, & \alpha_4^* = (\alpha_5s\pm\alpha_4x^2)x^2, \\ \alpha_5^* = \alpha_5x^6, & \alpha_6^* = \alpha_6x^4\pm2\alpha_7sx^2+\alpha_8s^2, \\ \alpha_7^* = (\alpha_8s\pm\alpha_7x^2)x^4, & \alpha_8^* = \alpha_8x^8. \end{array} |
We will consider only the action of
2.
3.
4.
\langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_8 \rangle; |
\langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_8 \rangle. |
Summarizing all cases we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+ \alpha\nabla_2 +\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_8 \rangle^{ { \begin{array}{l}O(\alpha,\beta,\gamma,\mu) = O(\pm \alpha,\pm \eta_5^4 \beta,\eta_5^2\gamma,\eta_5^4\mu) = \\ O(\pm \alpha,\mp \eta_5^3 \beta,\eta_5^4\gamma,-\eta_5^3\mu) = O(\pm \alpha,\pm \eta_5^2 \beta,-\eta_5\gamma,\eta_5^2\mu) = \\ O(\pm \alpha,\mp \eta_5 \beta,-\eta_5^3\gamma,-\eta_5\mu) \end{array}}}, \\ \langle \nabla_1+\nabla_3 +\alpha\nabla_4 + \beta\nabla_6\rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(-\alpha,\beta) = \\ O(\alpha,-\beta) = O(-\alpha,-\beta) \end{array}}}, \\ \langle \nabla_1+ \alpha\nabla_3 +\beta \nabla_4+\nabla_7 \rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(-\eta_3\alpha,\eta_3^2\beta) = \\ O(\eta_3^2\alpha,-\eta_3\beta) \end{array}}}, \\ \langle \nabla_2+ \alpha\nabla_3+\nabla_5 + \beta\nabla_6+\gamma\nabla_7 \rangle ^{{ \begin{array}{l} O(\alpha,\beta,\gamma) = O(-\alpha,\beta,-\gamma) = O(-\eta_3\alpha,\eta_3^2\beta,\gamma) = \\ O(\eta_3\alpha,\eta_3^2\beta,-\gamma) = O(\eta_3^2\alpha,-\eta_3\beta,\gamma) = O(-\eta_3^2\alpha,-\eta_3\beta,-\gamma) \end{array}}}, \\ \langle \nabla_2+ \alpha\nabla_4 +\beta\nabla_5 + \gamma\nabla_6+\nabla_8 \rangle ^{ { \begin{array}{l} O(\alpha,\beta,\gamma) = O(-\alpha,\beta,\gamma) = O(\eta_5^4 \alpha,\eta_5^2\beta,\eta_5^4\gamma) = \\ O(-\eta_5^4\alpha,\eta_5^2\beta,\eta_5^4\gamma) = O(-\eta_5^3\alpha,\eta_5^4\beta,-\eta_5^3\gamma) = \\ O(\eta_5^3\alpha,\eta_5^4\beta,-\eta_5^3\gamma) = O(\eta_5^2\alpha,-\eta_5\beta,\eta_5^2\gamma) = \\ O(-\eta_5^2\alpha,-\eta_5\beta,\eta_5^2\gamma) = O(-\eta_5\alpha,-\eta_5^3\beta,-\eta_5\gamma) = \\ O(\eta_5\alpha,-\eta_5^3\beta,-\eta_5\gamma) \end{array}}}, \\ \langle \nabla_3 \rangle, \, \langle \nabla_3+ \nabla_4 +\alpha\nabla_6 \rangle, \, \langle \nabla_3+ \alpha\nabla_4+\nabla_7 \rangle ^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_3+ \nabla_5+\alpha\nabla_6+\beta\nabla_7 \rangle^{O(\alpha,\beta) = O(\alpha,-\beta)}, \, \langle \nabla_3+\nabla_6 \rangle, \\ \langle \nabla_4+ \alpha\nabla_5+\beta\nabla_6+\nabla_8 \rangle^{{ \begin{array}{l} O(\alpha,\beta) = O(-i\alpha,-\beta) = \\ O(i\alpha,-\beta) = O(-\alpha,\beta) \end{array}}}, \\ \langle \nabla_4+\nabla_7 \rangle, \, \langle \nabla_5+\nabla_6+ \alpha\nabla_7 \rangle^{O(\alpha,\beta) = O(\alpha,-\beta)}, \\ \langle \nabla_5+\alpha\nabla_6+ \nabla_8 \rangle, \, \langle \nabla_5+ \alpha\nabla_7 \rangle^{O(\alpha) = O(-\alpha)}, \, \langle \nabla_6+ \nabla_8 \rangle, \, \langle \nabla_7 \rangle, \, \langle \nabla_8 \rangle,\end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{150}^{\alpha, \beta, \gamma, \mu } & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4+\mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{151}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4+\beta e_5 \\ {\mathbf{N}}_{152}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{153}^{\alpha, \beta, \gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = e_4+\beta e_5 & e_3e_4 = \gamma e_5 \\ {\mathbf{N}}_{154}^{\alpha, \beta, \gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4+\gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{155} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_3e_3 = e_4 \\ {\mathbf{N}}_{156}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_3 = e_5 & e_3e_3 = e_4+\alpha e_5 \\ {\mathbf{N}}_{157}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{158}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = e_4+\alpha e_5 & e_3e_4 = \beta e_5 \\ {\mathbf{N}}_{159} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_3e_3 = e_4+e_5 \\ {\mathbf{N}}_{160}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_3 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4+\beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{161} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_3 = e_5 \\&& e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{162}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\& & e_3e_3 = e_4+e_5 & e_3e_4 = \alpha e_5 \\ {\mathbf{N}}_{163}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\&& e_3e_3 = e_4+\alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{164}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\&& e_3e_3 = e_4 & e_3e_4 = \alpha e_5 \\ {\mathbf{N}}_{165} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4+e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{166} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{167} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|llll|} \hline {\mathbf{N}}^{4*}_{13}(\lambda) & \;\; e_1e_1 = e_2 \;\; e_1e_2 = e_3 \;\; e_1e_3 = e_4 \;\; e_2e_2 = \lambda e_4 \\ \hline & { \begin{array}{l} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{13}(2)) = \langle [\Delta_{22}],4[\Delta_{23}]+[\Delta_{14}],[\Delta_{24}]\rangle,\\\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{13}(2)) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{13}(2))\oplus \langle [\Delta_{23}], [\Delta_{33}], [\Delta_{34}], [\Delta_{44}]\rangle \\\mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{13}(\lambda )_{\lambda \neq2}) = \langle [\Delta_{22}],(3\lambda-2)[\Delta_{23}]+[\Delta_{14}]\rangle,\\\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{13}(\lambda)_{\lambda \neq2}) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{13}(\lambda)\oplus \langle [\Delta_{23}], [\Delta_{24}], [\Delta_{33}], [\Delta_{34}], [\Delta_{44}] \rangle \end{array} }\\ \hline &{ \phi = \begin{pmatrix} x&0&0&0\\ y&x^2&0&0\\ z&2xy&x^3&0\\ t&\lambda y^2+2xz&(\lambda+2)x^2y&x^4 \end{pmatrix} }\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{lllll} \nabla_1 = [\Delta_{14}]+(3\lambda-2)[\Delta_{23}], & \nabla_2 = [\Delta_{22}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&0&0&\alpha_1\\ 0&\alpha_2&(3\lambda-2)\alpha_1+\alpha_3&\alpha_4\\ 0&(3\lambda-2)\alpha_1+\alpha_3&\alpha_5&\alpha_6\\ \alpha_1&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^{**}&\alpha^{***}&\alpha^{*}&\alpha_1^*\\ \alpha^{***}&\alpha_2^*+\lambda\alpha^{*}&(3\lambda-2)\alpha^*_1+\alpha_3^*&\alpha^*_4\\ \alpha^{*}&(3\lambda-2)\alpha^*_1+\alpha_3^*&\alpha^*_5&\alpha^*_6\\ \alpha^*_1&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix}, |
we have
\begin{array}{lcl} \alpha_1^* = (\alpha_1x+\alpha_4y+\alpha_6z+\alpha_7t)x^4, \\ \alpha_2^* = \alpha_2x^4+4\lambda(\alpha_6y+\alpha_7z)xy^2+\lambda^2\alpha_7y^4+4(\alpha_4z+(\alpha_3+(3\lambda-2)\alpha_1)y)x^3 \\ \qquad\qquad{ +2(4\alpha_6yz+2\alpha_7z^2+(2\alpha_5+\lambda\alpha_4)y^2)x^2 }\\ \qquad{ -\lambda((\lambda+2)(\alpha_4y+\alpha_6z+\alpha_7t)y+((\alpha_3+4\lambda\alpha_1)y+\alpha_5z+\alpha_6t)x)x^2, }\\ \alpha_3^* = [(\lambda+2)(\alpha_4x^2+2\alpha_6xy+2\alpha_7xz+\lambda\alpha_7y^2)y \\\qquad { +((\alpha_3+(3\lambda-2)\alpha_1)x^2+2\alpha_5xy+2\alpha_6xz+\lambda\alpha_6y^2)x]x^2 }\\\qquad\qquad\qquad { -(3\lambda-2)(\alpha_1x+\alpha_4y+\alpha_6z+\alpha_7t)x^4, }\\ \alpha_4^* = (\alpha_4x^2+2\alpha_6xy+2\alpha_7xz+\lambda\alpha_7y^2)x^4, \\ \alpha_5^* = (\alpha_5x^2+2(\lambda+2)\alpha_6xy+(\lambda+2)^2\alpha_7y^2)x^4, \\ \alpha_6^* = (\alpha_6x+(\lambda+2)\alpha_7y)x^6, \\ \alpha_7^* = \alpha_7x^8. \end{array} |
We are interested in
(\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_7)\neq(0,0,0,0,0) {\text{ and }}(\alpha_1,\alpha_4,\alpha_6,\alpha_7)\neq(0,0,0,0). |
Let us consider the following cases:
\langle \alpha\nabla_1+\nabla_3 \rangle_{\alpha\notin \Big\{ 0,\frac{(\lambda-4)}{4(1-\lambda)(\lambda-2) }\Big\};\, \lambda\neq 1,2,4}; |
2.
x = 4 \alpha_4^2, y = -4 \alpha_1 \alpha_4, z = \alpha_1 \alpha_3 (4-\lambda)-\alpha_2 \alpha_4-\alpha_1^2 (8-12 \lambda+3 \lambda^2), t = 0, |
we have the representative
\begin{array}{c} x = \frac{\alpha_3+2(\lambda-2)\alpha_1}{\alpha_4}, y = -\frac{\alpha_1(\alpha_3+2(\lambda-2)\alpha_1)}{\alpha_4^2}, \\z = \frac{(2(2-\lambda)\alpha_1-\alpha_3)(\alpha_2\alpha_4+(\lambda-4)\alpha_1\alpha_3+(3\lambda^2-12\lambda+8)\alpha_1^2)}{4\alpha^3_4}, t = 0,\end{array} |
we have the representative
3.
x = \frac{\alpha_1}{\alpha_5}, y = -\frac{\alpha_1\alpha_3}{2\alpha_5^2}, z = \frac{\alpha_1(2\alpha_2\alpha_5+(\lambda-2)\alpha_3^2+4(\lambda^2-3\lambda+2)\alpha_1\alpha_3)}{2\lambda\alpha_5^3}, t = 0, |
we have the family of representatives
(b) if
x = 4 \alpha_4^3, y = -4 \alpha_1 \alpha_4^2, z = 4 \alpha_1 \alpha_3 \alpha_4-\alpha_2 \alpha_4^2-4 \alpha_1^2 (2 \alpha_4+\alpha_5), t = 0, |
we have the family of representatives
\begin{array}{c} x = \frac{\alpha_3\alpha_4-2\alpha_1(2\alpha_4+\alpha_5)}{\alpha_4\alpha_5)}, y = \frac{\alpha_1(2\alpha_1(2\alpha_4+\alpha_5)-\alpha_3\alpha_4)}{\alpha^2_4\alpha_5)},\\z = \frac{(2\alpha_1(2\alpha_4+\alpha_5)-\alpha_3\alpha_4)(\alpha_2\alpha_4^2-4\alpha_1\alpha_3\alpha_4+4\alpha_1^2(2\alpha_4+\alpha_5))}{4\alpha_4^4\alpha_5}, t = 0, \end{array} |
we have the family of representatives
(c) if
x = \frac{\lambda\alpha_3+2(\lambda^2-2\lambda-4)\alpha_1}{\lambda\alpha_5}, y = -\frac{4\alpha_1(\lambda\alpha_3+2(\lambda^2-2\lambda-4)\alpha_1)}{\lambda^2\alpha^2_5}, z = 0, t = 0, |
we have the family of representatives
\begin{array}{c} x = \frac{\sqrt{4\lambda(\lambda-4)\alpha_1\alpha_3+\lambda^2\alpha_2\alpha_5+4(3\lambda^3-12\lambda^2+8\lambda+16)\alpha_1^2}}{\lambda\alpha_5}, \\ y = -\frac{4\alpha_1\sqrt{4\lambda(\lambda-4)\alpha_1\alpha_3+\lambda^2\alpha_2\alpha_5+4(3\lambda^3-12\lambda^2+8\lambda+16)\alpha_1^2}}{\lambda^2\alpha^2_5}, z = 0, t = 0,\end{array} |
we have the family of representatives
y = -\frac{\alpha_1}{\alpha_4}x, z = -\frac{\alpha_2\alpha_4^2+(\lambda-4)\alpha_1\alpha_3\alpha_4+\alpha_1^2(4\alpha_5+(3\lambda^2-12\lambda+8)\alpha_4)}{\alpha^2_4(4\alpha_4-\lambda\alpha_5)}x, t = 0, |
we have two families of representatives
\langle \alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq\frac{\lambda}{4}} {\text{ and }} \langle \nabla_3+\alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq\frac{\lambda}{4}} |
depending on
4.
5.
\begin{array}{c} y = -\frac{\alpha_6}{\alpha_7(\lambda+2)}x, z = \frac{2(\lambda+2)^2\alpha_4\alpha_7-(\lambda+4)\alpha_6^2}{2(\lambda+2)^2\alpha^2_7}x,\\ t = \frac{(\lambda^2+6\lambda+8)\alpha_4\alpha_6\alpha_7-2(\lambda+2)^2\alpha_1\alpha^2_7-(\lambda+4)\alpha^3_6}{2(\lambda+2)^2\alpha^3_7}x ,\end{array} |
we have
6.
(b) if
x = \sqrt{\frac{\alpha_5}{\alpha_7}}, y = \frac{8\alpha_1-\alpha_3}{2\sqrt{\alpha_5\alpha_7}}, z = \frac{\alpha_7(\alpha_3-8\alpha_1)^2-2\alpha_4\alpha_5^2}{4\alpha_5\alpha_7\sqrt{\alpha_5\alpha_7}}, t = \frac{\alpha_3\alpha_4-2\alpha_1(4\alpha_4+\alpha_5)}{2\alpha_7\sqrt{\alpha_5\alpha_7}}, |
we have the family of representatives
(c) if
x = \frac{\alpha_6}{\alpha_7}, y = 0, z = -\frac{\alpha_4\alpha_6}{2\alpha_7^2}, t = \frac{\alpha_6(\alpha_4\alpha_6-2\alpha_1\alpha_7)}{2\alpha_7^3}, |
we have the family of representatives
\begin{array}{c} x = \frac{\alpha_6}{\alpha_7}, y = \frac{\alpha_6 (\alpha_2 \alpha_7-\alpha_4^2 - 2 \alpha_1 \alpha_6 )}{ 6 \alpha_7 (\alpha_4 \alpha_6 + 8 \alpha_1 \alpha_7 - \alpha_3 \alpha_7))},\\ z = \frac{y^2}{x}-\frac{\alpha_4x}{2\alpha_7}-\frac{\alpha_6y}{\alpha_7}, t = -\frac{x \alpha_1 + y \alpha_4 + z \alpha_6}{\alpha_7},\end{array} |
we have the family of representatives
\langle \alpha\nabla_3+\nabla_5+\nabla_6+\nabla_7 \rangle_{\alpha\neq0,\lambda = -2}; |
\begin{array}{c} x = \frac{\alpha_6}{\alpha_7}, y = \frac{\alpha_6 (\alpha_4 \alpha_6+8 \alpha_1 \alpha_7-\alpha_3 \alpha_7)}{2 \alpha_7 (-\alpha_6^2+\alpha_5 \alpha_7)}, z = \frac{y^2}{x}-\frac{\alpha_4}{2\alpha_7}x-\frac{\alpha_6}{\alpha_7}y,\\t = \frac{(\alpha_4\alpha_6-2\alpha_1\alpha_7)x^2-2\alpha_6\alpha_7y^2+2(\alpha_6^2-\alpha_4\alpha_7)xy}{2\alpha_7^2x},\end{array} |
we have the family of representatives
\langle \alpha\nabla_2+\beta\nabla_5+\nabla_6+\nabla_7 \rangle_{\beta\neq1,\lambda = -2}, |
which will be jointed with the family from the case (6(c)i).
Summarizing all cases we have the following distinct orbits:
\begin{array}{c} \langle (\lambda-4)\nabla_1+4(1-\lambda)(\lambda-2)(\nabla_2+\nabla_3) \rangle_{\lambda \notin \{ 1; 2; 4 \}}, \langle \nabla_1+\alpha\nabla_2+\nabla_5 \rangle_{\lambda = 0, \alpha \neq 0}, \\ \langle \alpha\nabla_1+\nabla_3 \rangle_{\alpha\neq0}, \, \langle \nabla_1+\nabla_5 \rangle, \langle \nabla_2+\alpha\nabla_3+\frac{\lambda}{4}\nabla_4+\nabla_5 \rangle_{\lambda\neq0}^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_2+\alpha\nabla_3+\beta\nabla_5+\nabla_6 \rangle _{\lambda = 0}^{O(\alpha,\beta) = O(\eta_3^2\alpha,-\eta\beta) = O(-\eta_3\alpha,\eta_3^2\beta)}, \\ \langle \alpha\nabla_2+\nabla_3+ \nabla_7 \rangle_{\lambda\neq-2}^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \alpha\nabla_2+\beta\nabla_5+\nabla_6+\nabla_7 \rangle_{\lambda = -2}, \\ \langle \alpha\nabla_2+\nabla_5+\nabla_7 \rangle, \langle \nabla_2 + \nabla_7 \rangle, \langle \nabla_3+\nabla_4 \rangle, \langle \nabla_3+\alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq0}, \\ \langle \nabla_3+\alpha\nabla_5+\nabla_6 \rangle, \langle \alpha\nabla_3+\nabla_5+\nabla_6+\nabla_7 \rangle_{\alpha\neq0,\lambda = -2}, \langle \nabla_3+\nabla_7 \rangle_{\lambda = -2}, \langle \nabla_4 \rangle_{\lambda\neq 2}, \\ \langle \alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq0}, \langle \nabla_5+\nabla_6 \rangle, \langle \nabla_6 \rangle, \langle \nabla_7 \rangle. \end{array} |
Now we have the following new algebras
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{168}^{\lambda \neq 1; 2; 4} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = (\lambda-4)e_5 \\ & & { e_2e_2 = \lambda e_4 + 4(1-\lambda)(\lambda-2)e_5 } & { e_2e_3 = - \lambda(\lambda+2)e_5 }\\ {\mathbf{N}}_{169}^{\alpha\neq0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = -2 e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{170}^{\lambda, \alpha \neq 0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_1e_4 = \alpha e_5 & e_2e_2 = \lambda e_4 & { e_2e_3 = (1+\alpha(3\lambda-2)) e_5 } \\ {\mathbf{N}}_{171}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = \lambda e_4 & { e_2e_3 = (3\lambda-2) e_5 } & e_3e_3 = e_5 \\ {\mathbf{N}}_{172}^{\lambda \neq 0,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 + e_5 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = \frac{\lambda} {4} e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{173}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{174}^{\lambda\neq-2, \alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 + \alpha e_5 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{175}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = -2 e_4+\alpha e_5 \\ && e_3e_3 = \beta e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{176}^{\lambda,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4+\alpha e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{177}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4+ e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{178}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 \\ {\mathbf{N}}_{179}^{\lambda,\alpha\neq0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{180}^{\lambda,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{181}^{\alpha \neq 0 } & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = -2 e_4 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{182} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_2 = -2 e_4 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{183}^{\lambda\neq 2} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_2 = \lambda e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{184}^{\lambda, \alpha\neq 0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{185}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_2 = \lambda e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{186}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{187}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4 & e_4e_4 = e_5 \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{14} & \begin{array}{l} e_1e_2 = e_3 \\ e_1e_3 = e_4 \end{array} & \begin{array}{lcl} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{14})& = &\\ {\langle [\Delta_{11}],[\Delta_{22}],[\Delta_{23}],[\Delta_{33}]\rangle}\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{14})& = &\mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{14})\oplus\\ {\langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}]\rangle } \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&q&0&0\\ 0&r&xq&0\\ t&s&xr&x^2q \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} \alpha_1&0&0&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ 0&\alpha_4&\alpha_6&\alpha_7\\ \alpha_2&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{**}&\alpha_2^*\\ \alpha^{*}&\alpha_3^*&\alpha^*_4&\alpha^*_5\\ \alpha^{**}&\alpha^*_4&\alpha^*_6&\alpha^*_7\\ \alpha^*_2&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix}, |
we have
\begin{array}{lcl} \alpha_1^* & = & \alpha_1x^2+2\alpha_2xt+\alpha_8t^2, \\ \alpha_2^* & = & (\alpha_2x+\alpha_8t)x^2q, \\ \alpha_3^* & = & (\alpha_3q+\alpha_4r+\alpha_5s)q+(\alpha_4q+\alpha_6r+\alpha_7s)r+(\alpha_5q+\alpha_7r+\alpha_8s)s, \\ \alpha_4^* & = & (\alpha_4q+\alpha_6r+\alpha_7s)xq+(\alpha_5q+\alpha_7r+\alpha_8s)xr, \\ \alpha_5^* & = & (\alpha_5q+\alpha_7r+\alpha_8s)x^2q, \\ \alpha_6^* & = & (\alpha_6q^2+2\alpha_7qr+\alpha_8r^2)x^2, \\ \alpha_7^* & = & (\alpha_7q+\alpha_8r)x^3q, \\ \alpha_8^* & = & \alpha_8x^4q^2. \end{array} |
We are interested in
\begin{array}{c} x = \frac{\sqrt{\alpha_3\alpha_6-\alpha_4^2}}{\alpha_6},q = \frac{\alpha_2\sqrt{\alpha_3\alpha_6-\alpha_4^2}}{\alpha^2_6},r = -\frac{\alpha_2\alpha_4\sqrt{\alpha_3\alpha_6-\alpha_4^2}}{\alpha^3_6},\\ s = 0, t = -\frac{\alpha_1\sqrt{\alpha_3\alpha_6-\alpha_4^2}}{2\alpha_2\alpha_6}, \end{array} |
we have the representative
2.
x = 1, r = -\frac{\alpha_4}{\alpha_5}q, s = \frac{2\alpha_4^2-\alpha_3\alpha_5}{2\alpha_5^2}q, t = 0, |
we have the representatives
x = \alpha_5, q = \alpha_2, r = -\frac{\alpha_2\alpha_4}{\alpha_5}, s = \frac{\alpha_2(2\alpha_4^2-\alpha_3\alpha_5)}{2\alpha_5^2}, t = -\frac{\alpha_1\alpha_5}{2\alpha_2}, |
we have the representatives
x = 1, q = \sqrt{\frac{\alpha_1}{\alpha_5}}, r = 0, s = -\frac{\alpha_3\sqrt{\alpha_1}}{2\alpha_5\sqrt{\alpha_5}}, t = 0, |
we have the representative
x = {\frac{\alpha_4}{\alpha_5}}, q = 1, r = 0, s = -\frac{\alpha_3}{2\alpha_5}, t = 0, |
we have the representative
x = \frac{\alpha_4}{\alpha_5}, q = \sqrt{\frac{\alpha_1}{\alpha_5}}, r = 0, s = -\frac{\alpha_3\sqrt{\alpha_1}}{2\alpha_5\sqrt{\alpha_5}}, t = 0, |
we have the representative
x = \alpha_5, q = \alpha_2, r = 0, s = -\frac{\alpha_2\alpha_3}{2\alpha_5}, t = -\frac{\alpha_1\alpha_5}{2\alpha_2}, |
we have the representative
x = \frac{\alpha_4}{\alpha_5}, q = {\frac{\alpha_2\alpha_4}{\alpha^2_5}}, r = 0, s = -\frac{\alpha_2\alpha_3\alpha_4}{2\alpha_5^3}, t = -\frac{\alpha_1\alpha_4}{2\alpha_2\alpha_5}, |
we have the representative
(d) if
x = 1, q = 1, s = \frac{ \alpha_4^2 (2 \alpha_5+\alpha_6)-\alpha_3 (\alpha_5+\alpha_6)^2}{2 \alpha_5 (\alpha_5+\alpha_6)^2}, r = -\frac{\alpha_4}{\alpha_5+\alpha_6}, t = 0, |
we have the family of representatives
\begin{array}{c} x = 1, q = \sqrt{\frac{\alpha_1}{\alpha_5}}, r = -\frac{\alpha_4\sqrt{\alpha_1}}{(\alpha_5+\alpha_6)\sqrt{\alpha_5}}, \\s = \frac{(\alpha_4^2(2\alpha_5+\alpha_6)-\alpha_3(\alpha_5+\alpha_6)^2)\sqrt{\alpha_1}}{2\alpha_5(\alpha_5+\alpha_6)^2\sqrt{\alpha_5}}, t = 0, \end{array} |
we have the family of representatives
x = \alpha_5, q = \alpha_2, r = -\frac{\alpha_2\alpha_4}{\alpha_5+\alpha_6}, s = \frac{\alpha_2(\alpha_4^2(2\alpha_5+\alpha_6)-\alpha_3(\alpha_5+\alpha_6)^2)}{2\alpha_5(\alpha_5+\alpha_6)^2}, t = -\frac{\alpha_1\alpha_5}{2\alpha_2}, |
we have the family of representatives
3.
x = \sqrt[3]{\frac{\alpha_3}{\alpha_7}}, q = \sqrt[6]{\frac{\alpha^3_1}{\alpha_3\alpha^2_7}}, r = 0, s = 0, t = 0, |
we have the family of representatives
4.
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+ \nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle^{O(\alpha,\beta) = O(i\alpha,-\beta) = O(-i\alpha,-\beta) = O(-\alpha,\beta)}, \\ \langle \nabla_1+\nabla_3+\alpha\nabla_6 +\nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_1+ \nabla_4+\nabla_5-\nabla_6 \rangle, \\ \langle \nabla_1+ \nabla_4+\alpha\nabla_6+\nabla_8 \rangle^{O(\alpha) = O(-\eta_3 \alpha) = O(\eta_3^2 \alpha)}, \langle \nabla_1+\nabla_5+\alpha\nabla_6 \rangle, \langle \nabla_1+\nabla_6 +\nabla_7 \rangle, \\ \langle \nabla_1+\nabla_6+\nabla_8 \rangle, \langle \nabla_1+\nabla_7 \rangle, \langle \nabla_1+\nabla_8 \rangle, \langle \nabla_2 \rangle, \langle \nabla_2+\nabla_3 \rangle, \langle \nabla_2+ \nabla_3+\nabla_6 \rangle, \\ \langle \nabla_2+\nabla_3+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_2+\nabla_4 \rangle, \langle \nabla_2+\nabla_4+\nabla_5-\nabla_6 \rangle, \\ \langle \nabla_2+\nabla_5+\alpha\nabla_6 \rangle, \langle \nabla_2+\nabla_6 \rangle, \langle \nabla_2+ \nabla_6+\nabla_7 \rangle, \langle \nabla_2+\nabla_7 \rangle, \\ \langle \nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle^{O(\alpha,\beta) = O(i\alpha,-\beta) = O(-i\alpha,-\beta) = O(-\alpha,\beta)}, \\ \langle \nabla_3+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\eta_3 \alpha) = O(\eta_3^2 \alpha)}, \langle \nabla_4+\nabla_5-\nabla_6 \rangle, \\ \langle \nabla_4+\alpha\nabla_6+\nabla_8 \rangle^{O(\alpha) = O(-\eta_3 \alpha) = O(\eta_3^2 \alpha)}, \langle \nabla_5+\alpha\nabla_6 \rangle, \langle \nabla_6+\nabla_7 \rangle, \langle \nabla_6+\nabla_8 \rangle, \\ \langle \nabla_7 \rangle, \langle \nabla_8 \rangle, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{188}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{189}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{190} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{191}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{191}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{192} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{193} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{194} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{195} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{196} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ {\mathbf{N}}_{197} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ {\mathbf{N}}_{198} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{199}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\&& e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{200} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{201} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\&& e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{202}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{203} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{204} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{205} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{206}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\&& e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{207}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{208} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{209}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{210}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{211} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{212} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{213} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{214} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{15} & \begin{array}{l} e_1e_2 = e_3 \\ e_1e_3 = e_4 \\ e_2e_2 = e_4 \end{array} & \begin{array}{lcl} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{15})& = &\\ {\langle [\Delta_{11}],[\Delta_{22}],[\Delta_{23}],[\Delta_{33}]\rangle}\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{15}) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{15})\oplus\\ {\langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle } \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&r&x^3&0\\ t&s&xr&x^4 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} \alpha_1&0&0&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ 0&\alpha_4&\alpha_6&\alpha_7\\ \alpha_2&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{**}&\alpha_2^*\\ \alpha^{*}&\alpha_3^*+\alpha^{**}&\alpha^*_4&\alpha^*_5\\ \alpha^{**}&\alpha^*_4&\alpha^*_6&\alpha^*_7\\ \alpha^*_2&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix}, |
we have
\begin{array}{lcl} \alpha_1^* & = & \alpha_1x^2+2\alpha_2xt+\alpha_8t^2, \\ \alpha_2^* & = & (\alpha_2x+\alpha_8t)x^4, \\ \alpha_3^* & = & x^4 \alpha_3+2 r x^2 \alpha_4+2 s x^2 \alpha_5+r^2 \alpha_6+2 r s \alpha_7+ \\ &&{ s^2 \alpha_8-x (r x \alpha_2+t x^2 \alpha_7+r t \alpha_8), }\\ \alpha_4^* & = & (\alpha_4x^2+\alpha_6r+\alpha_7s)x^3+(\alpha_5x^2+\alpha_7r+\alpha_8s)xr, \\ \alpha_5^* & = & (\alpha_5x^2+\alpha_7r+\alpha_8s)x^4, \\ \alpha_6^* & = & (\alpha_6x^4+2\alpha_7x^2r+\alpha_8r^2)x^2, \\ \alpha_7^* & = & (\alpha_7x^2+\alpha_8r)x^5, \\ \alpha_8^* & = & \alpha_8x^8. \end{array} |
We are interested in
2.
\begin{array}{c} x = 2 \alpha_5 (\alpha_5+\alpha_6), s = 2 \alpha_5 (\alpha_4^2 (2 \alpha_5+\alpha_6)-\alpha_3 (\alpha_5+\alpha_6)^2),\\ r = -4 \alpha_4 \alpha_5^2 (\alpha_5+\alpha_6),\end{array} |
we have the family of representatives
x = \sqrt[4]{\frac{\alpha_1}{\alpha_5}}, r = -\frac{\alpha_4\sqrt{\alpha_1}}{(\alpha_5+\alpha_6)\sqrt{\alpha_5}}, s = \frac{((\alpha_5+\alpha_6)(2\alpha_4^2-\alpha_2\alpha_4)-\alpha_3(\alpha_5+\alpha_6)^2-\alpha_4^2\alpha_6)\sqrt{\alpha_1}}{2\alpha_5(\alpha_5+\alpha_6)^2\sqrt{\alpha_5}}, t = 0, |
we have the family of representatives
\begin{array}{c} x = \frac{\alpha_2}{\alpha_5}, r = -\frac{\alpha_2^2\alpha_4}{\alpha_5^2(\alpha_5+\alpha_6)}, \\s = \frac{\alpha_2^2((\alpha_5+\alpha_6)(2\alpha_4^2-\alpha_2\alpha_4)-\alpha_3(\alpha_5+\alpha_6)^2-\alpha_4^2\alpha_6)}{2\alpha^3_5(\alpha_5+\alpha_6)^2}, t = -\frac{\alpha_1}{2\alpha_5}, \end{array} |
we have the family of representatives
(b) if
\langle \alpha\nabla_1+\nabla_4+\nabla_5-\nabla_6 \rangle {\text{ and }} \langle \alpha\nabla_2+\nabla_4+\nabla_5-\nabla_6 \rangle_{\alpha\neq0} |
depending on
3.
r = -\frac{\alpha_5}{\alpha_7}x^2, s = \frac{\alpha_5\alpha_6-\alpha_4\alpha_7}{\alpha_7^2}x^2, t = \frac{\alpha_3\alpha_7^2-2\alpha_4\alpha_5\alpha_7+\alpha_5^2\alpha_6+\alpha_2\alpha_5\alpha_7}{\alpha_7^3}x, |
we have
4.
Summarizing all cases we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\beta\nabla_6+\nabla_7 \rangle ^{O(\alpha,\beta) = O(\eta_5^2\alpha,-\eta_5\beta) = O(\eta_5^4\alpha,\eta_5^2\beta) = O(-\eta_5\alpha,-\eta_5^3\beta) = O(-\eta_5^3\alpha,\eta_5^4\beta)}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4 +\gamma\nabla_6+\nabla_8 \rangle^{ { \begin{array}{l}O(\alpha,\beta,\gamma) = O(-\eta_3\alpha,\beta,\eta_3^2\gamma) = O(-\eta_3\alpha,-\beta,\eta_3^2\gamma) = \\ O(\eta_3^2\alpha,-\beta,-\eta_3\gamma) = O(\eta_3^2\alpha,\beta,-\eta_3\gamma) = O(\alpha,-\beta,\gamma) \end{array}}}, \\ \langle \alpha\nabla_1+ \nabla_4+\nabla_5-\nabla_6 \rangle, \langle \nabla_1+\nabla_5+\alpha\nabla_6 \rangle, \langle 2\nabla_2+\nabla_3 +\nabla_4 \rangle, \langle \nabla_2+\alpha\nabla_3+\nabla_6 \rangle, \\ \langle \nabla_2+\alpha\nabla_4 \rangle, \langle \alpha\nabla_2+\nabla_4+\nabla_5-\nabla_6 \rangle_{\alpha\neq0}, \langle \nabla_2+\nabla_5+\alpha\nabla_6 \rangle, \\ \langle \nabla_2+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle ^{O(\alpha,\beta) = O(i\alpha,-\beta) = O(-i\alpha,-\beta) = O(-\alpha,\beta)}, \\ \langle \nabla_4+\alpha\nabla_6+\nabla_8 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_5+\alpha\nabla_6 \rangle, \langle \nabla_6+ \nabla_7 \rangle, \langle \nabla_6+\nabla_8 \rangle, \\ \langle \nabla_7 \rangle, \langle \nabla_8 \rangle, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{215}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{216}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4+\alpha e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{217}^{\alpha} & : & e_1e_1 = \alpha e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{218}^\alpha & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{219} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = 2e_5 & e_2e_2 = e_4+e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{220}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_4+\alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{221}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{222}^{\alpha\neq 0} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{223}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\&& e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{224}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\&& e_2e_2 = e_4 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{225}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & { e_2e_2 = e_4+e_5 } \\&& e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{226}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\&& e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{227}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{228} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{229} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{230} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{231} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} \alpha_1&0&0&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ 0&\alpha_4&\alpha_6&\alpha_7\\ \alpha_2&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{**}&\alpha_2^*\\ \alpha^{*}&\alpha^*_3&\alpha^*_4+\alpha^{**}&\alpha_5^*\\ \alpha^{**}&\alpha^*_4+\alpha^{**}&\alpha_6^*&\alpha^*_7\\ \alpha^*_2&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix}, |
in the case
\begin{array}{ll} \alpha_1^* = \alpha_1x^2+2\alpha_2xt+\alpha_8t^2, & \alpha_2^* = (\alpha_2x+\alpha_8t)x^3, \\ \alpha_3^* = \alpha_3x^2+2\alpha_5xs+\alpha_8s^2, & \alpha_4^* = (\alpha_4x+\alpha_7s)x^2-\alpha_7x^2t, \\ \alpha_5^* = (\alpha_5x+\alpha_8s)x^3, & \alpha_6^* = \alpha_6x^4, \\ \alpha_7^* = \alpha_7x^5, & \alpha_8^* = \alpha_8x^6; \end{array} |
and on the opposite case, for
\begin{array}{ll} \alpha_1^* = \alpha_3y^2+2 \alpha_5 t y+ \alpha_8 t^2, & \alpha_2^* = (\alpha_5 y + \alpha_8t )y^3, \\ \alpha_3^* = \alpha_1y^2+2 \alpha_2s y + \alpha_8s^2, & \alpha_4^* = ((s-t) \alpha_7-y \alpha_4)y^2, \\ \alpha_5^* = (y \alpha_2+s \alpha_8)y^3, & \alpha_6^* = \alpha_6 y^4, \\ \alpha_7^* = \alpha_7y^5, & \alpha_8^* = \alpha_8 y^6. \end{array} |
We are interested in
2.
\phi = \phi_1, x = \frac{\alpha_4}{\alpha_5}, t = -\frac{\alpha_1 \alpha_4}{ 2 \alpha_2 \alpha_5}, s = -\frac{ \alpha_3 \alpha_4}{ 2 \alpha_5^2}, |
we have the following family of representatives
\langle \alpha \nabla_2+\nabla_4+\nabla_5+ \beta \nabla_6 \rangle_{\alpha\neq0}; |
\phi = \phi_1, x = 2 \alpha_2 \alpha_5, t = - \alpha_1 \alpha_5, s = - \alpha_2 \alpha_3, |
we have the following family of representatives
3.
\phi = \phi_1, x = \sqrt[3]{{\alpha_1}{\alpha_7^{-1}}}, s = 0, t = {\alpha_4\sqrt[3]{\alpha_1} \alpha_7^{-1}}, |
we have the family of representatives
\phi = \phi_1, x = {\alpha_2}{\alpha_7^{-1}}, s = -({\alpha_1\alpha_7+2\alpha_2\alpha_4}) /(2\alpha_7^2), t = -{\alpha_1}/ ({2\alpha_7}), |
we have the family of representatives
4.
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_6+\mu\nabla_7+\nabla_8\rangle ^{ { \begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(\alpha,i\beta,-\gamma,-i\mu) = \\ O(\alpha,-i\beta,-\gamma,i\mu) = O(\alpha,-\beta,\gamma,-\mu) = \\ O(\frac{1}{\alpha},-\frac{\beta}{\sqrt[4]{\alpha^{3}}}, \frac{\gamma}{\sqrt{\alpha}},\frac{\mu}{\sqrt[4]{\alpha}}) = \\ O(\frac{1}{\alpha},-\frac{i\beta}{ \sqrt[4]{\alpha^{3}}},-\frac{\gamma}{\sqrt{\alpha}},-\frac{i\mu}{\sqrt[4]{\alpha}}) = \\ O(\frac{1}{\alpha},\frac{\beta}{\sqrt[4]{\alpha^{3}}}, \frac{\gamma}{\sqrt{\alpha}},-\frac{\mu}{\sqrt[4]{\alpha}}) = O(\frac{1}{\alpha},-\frac{\beta}{\sqrt[4]{\alpha^{3}}}, \frac{\gamma}{\sqrt{\alpha}},\frac{\mu}{\sqrt[4]{\alpha}})\end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_6+\nabla_7\rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(\alpha,-\eta_3\beta) = O(\alpha,\eta_3^2\beta) = \\ O(\alpha^{-1},-\eta_3\beta\sqrt[3]{\alpha^{-1}}) = O(\alpha^{-1},\eta_3^2\beta\sqrt[3]{\alpha^{-1}}) = O(\alpha^{-1},\beta\sqrt[3]{\alpha^{-1}})\end{array}}}, \\ \langle \nabla_2+\alpha\nabla_3+\nabla_4 +\beta \nabla_6 \rangle, \langle \nabla_2+\alpha\nabla_3+\beta\nabla_5+\gamma\nabla_6+\nabla_7\rangle^{O(\alpha,\beta,\gamma) = O(-\frac{\alpha}{\beta^4},\frac{1}{\beta},\frac{\gamma}{\beta})}, \\ \langle \nabla_2+\nabla_3+\alpha \nabla_6 \rangle, \langle \alpha \nabla_2+\nabla_4+\nabla_5+ \beta \nabla_6 \rangle_{\alpha\neq0}^{O(\alpha,\beta) = O(\alpha^{-1},\beta\alpha^{-1})}, \\ \langle \alpha \nabla_2+ \nabla_5+ \beta \nabla_6 \rangle_{\alpha\neq0}^{O(\alpha,\beta) = O(\alpha^{-1},\beta\alpha^{-1})}, \langle \nabla_2+ \alpha \nabla_6 \rangle, \\ \langle \nabla_4+\alpha\nabla_6+\beta\nabla_7+\nabla_8\rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(\eta_3^2\alpha,-\eta_3\beta) = O(-\eta_3\alpha,\eta_3^2\beta) = \\ O(\eta_3^2\alpha,\eta_3\beta) = O(-\eta_3\alpha,-\eta_3^2\beta) = O(\alpha,-\beta)\end{array} }}, \langle \nabla_6+\nabla_7\rangle, \\ \langle \nabla_6+\alpha\nabla_7+\nabla_8\rangle^{O(\alpha) = O(-\alpha)}, \langle \nabla_7 \rangle, \langle \nabla_7+\nabla_8\rangle, \langle \nabla_8\rangle, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{232}^{\alpha, \beta, \gamma, \mu} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4+\beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{233}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{234}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = e_4+e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{235}^{\alpha, \beta, \gamma} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{236}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{237}^{\alpha\neq0, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_3 = e_4+e_5 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{238}^{\alpha\neq0, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{239}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{240}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4+e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{241} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{242}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{243} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{244} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{245} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{22}], \\ \nabla_5 = [\Delta_{23}], & \nabla_6 = [\Delta_{24}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} \alpha_1&0&\alpha_2&\alpha_3\\ 0&\alpha_4&\alpha_5&\alpha_6\\ \alpha_2&\alpha_5&0&\alpha_7\\ \alpha_3&\alpha_6&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{*}_2&\alpha_3^*\\ \alpha^{*}&\alpha^*_4&\alpha^*_5&\alpha_6^*\\ \alpha^{*}_2&\alpha^*_5& 0 &\alpha^*_7\\ \alpha^*_3&\alpha^*_6&\alpha^*_7&\alpha^*_8 \end{pmatrix}, |
then in the case
\begin{array}{ll} \alpha_1^* = \alpha_1x^2+2\alpha_3xt+\alpha_8t^2, & \alpha_2^* = (\alpha_2x+\alpha_7t)xq, \\ \alpha_3^* = (\alpha_3x+\alpha_8t)x^2q^2, & \alpha_4^* = \alpha_4q^2+2\alpha_6qs+\alpha_8s^2, \\ \alpha_5^* = (\alpha_5q+\alpha_7s)xq, & \alpha_6^* = (\alpha_6q+\alpha_8s)x^2q^2, \\ \alpha_7^* = \alpha_7x^3q^3, & \alpha_8^* = \alpha_8x^4q^4; \end{array} |
and in the opposite case
\begin{array}{ll} \alpha_1^* = \alpha_4p^2+2 \alpha_6p t+ \alpha_8t^2, & \alpha_2^* = ( \alpha_5p + \alpha_7t)p y, \\ \alpha_3^* = ( \alpha_6p+ \alpha_8t)p^2 y^2 , & \alpha_4^* = \alpha_1y^2+2 \alpha_3s y+ \alpha_8s^2, \\ \alpha_5^* = ( \alpha_2y+ \alpha_7s)p y, & \alpha_6^* = ( \alpha_3y+ \alpha_8s)p^2 y^2 , \\ \alpha_7^* = \alpha_7p^3 y^3, & \alpha_8^* = \alpha_8p^4 y^4. \end{array} |
We are interested in
2.
3.
\phi = \phi_1, x = \sqrt[3]{{\alpha_4}{\alpha_3^{-1}}}, q = {\alpha_3}{\alpha_7^{-1}}, s = 0, t = 0, |
we have the family of representatives
\phi = \phi_1, x = \sqrt[8]{{\alpha^3_4}{\alpha_1^{-1}\alpha_7^{-2}}}, q = \sqrt[8]{{\alpha_1^3}{\alpha_4^{-1}\alpha^{-2}_7}}, s = 0, t = 0, |
we have the family of representatives
4.
\phi = \phi_1, x = \sqrt[5]{{\alpha^3_5}{\alpha_2^{-2}\alpha_8^{-1}}}, q = \sqrt[5]{{\alpha^3_2}{\alpha_5^{-2}\alpha_8^{-1}}}, s = 0, t = 0, |
we have the family of representatives
\phi = \phi_1, x = {\alpha_4}{\alpha_5^{-1}}, q = {\alpha^2_5}{\alpha_4^{-1}\sqrt{\alpha_4^{-1}\alpha_8^{-1}}}, s = 0, t = 0, |
we have the family of representatives
\phi = \phi_1, x = \sqrt[8]{{\alpha_4^3}{\alpha_2^{-2}\alpha_8^{-1}}}, q = \sqrt[4]{{\alpha^2_2}{\alpha_4^{-1}\alpha_8^{-1}}}, s = 0, t = 0, |
we have the family of representatives
\phi = \phi_1, x = \sqrt[6]{{\alpha^2_4}{\alpha_1^{-1}\alpha_8^{-1}}}, q = \sqrt[6]{{\alpha_1^2}{\alpha_4^{-1}\alpha_8^{-1}}}, s = 0, t = 0, |
we have the family of representatives
\langle \nabla_1+\alpha\nabla_2+\nabla_4+\beta\nabla_5+\gamma\nabla_7+\nabla_8 \rangle. |
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\nabla_4+\beta\nabla_5+\gamma\nabla_7+\nabla_8 \rangle^{ { \begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_3^2\alpha, \eta_3^2\beta, \eta_3^2\gamma) = O(-\eta_3^2\alpha, \eta_3^2\beta, -\eta_3^2\gamma) = \\ O(\eta_3^2\alpha, -\eta_3^2\beta, -\eta_3^2\gamma) = O(-\eta_3^2\alpha, \eta_3^2\beta, \eta_3^2\gamma) = \\ O(\eta_3\alpha, \eta_3\beta, -\eta_3\gamma) = O(-\eta_3\alpha, \eta_3\beta, \eta_3\gamma) = \\ O(\eta_3\alpha, -\eta_3\beta, \eta_3\gamma) = O(-\eta_3\alpha, -\eta_3\beta, -\eta_3\gamma) = \\ O(-\alpha, \beta, - \gamma) = O(\alpha, -\beta, - \gamma) = \\ O(-\alpha, -\beta, \gamma) = O(\beta,\alpha,\gamma) = \\ O(\eta_3^2\beta, \eta_3^2\alpha, \eta_3^2\gamma) = O(-\eta_3^2\beta, \eta_3^2\alpha, -\eta_3^2\gamma) = \\ O(\eta_3^2\beta, -\eta_3^2\alpha, -\eta_3^2\gamma) = O(-\eta_3^2\beta, \eta_3^2\alpha, \eta_3^2\gamma) = \\ O(\eta_3\beta, \eta_3\alpha, -\eta_3\gamma) = O(-\eta_3\beta, \eta_3\alpha, \eta_3\gamma) = \\ O(\eta_3\beta, -\eta_3\alpha, \eta_3\gamma) = O(-\eta_3\beta, -\eta_3\alpha, -\eta_3\gamma) = \\ O(-\beta, \alpha, - \gamma) = O(\beta, -\alpha, - \gamma) = O(-\beta, -\alpha, \gamma) \end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\nabla_4+\beta\nabla_6 +\nabla_7 \rangle^{ {\begin{array}{l} O(\alpha,\beta) = O(\eta_4\alpha,-\eta_4\beta) = O(-\eta_4\alpha,\eta_4\beta) = O(\eta_4^3\alpha,-\eta_4^3\beta) = \\ O(-\eta_4^3\alpha,\eta_4^3\beta) = O(-i\alpha,-i\beta) = O(i\alpha,i\beta) = O(-\alpha,-\beta) = \\ O(\beta,\alpha) = O(\eta_4\beta,-\eta_4\alpha) = O(-\eta_4\beta,\eta_4\alpha) = O(\eta_4^3\beta,-\eta_4^3\alpha) = \\ O(-\eta_4^3\beta,\eta_4^3\alpha) = O(-i\beta,-i\alpha) = O(i\beta,i\alpha) = O(-\beta,-\alpha) \end{array}}}, \\ \langle \nabla_2+\nabla_3 \rangle, \langle \nabla_2+\nabla_3+\nabla_4+\alpha\nabla_5 \rangle^{O(\alpha) = O(-\eta_3 \alpha) = O(\eta_3^2 \alpha)}, \\ \langle \nabla_2+\nabla_3+\nabla_5 \rangle, \langle \nabla_2+\nabla_3 +\alpha\nabla_5+\nabla_6 \rangle^{O(\alpha) = O(\alpha^{-1})}, \\ \langle \nabla_2+\nabla_4+\alpha\nabla_5+\beta\nabla_7+\nabla_8 \rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(\eta_4^3\alpha,-\eta_4^3\beta) = O(-\eta_4^3\alpha,\eta_4^3\beta) = O(\eta_4\alpha,-\eta_4\beta) = \\ O(-\eta_4\alpha,\eta_4\beta) = O(i\alpha,i\beta) = O(-i\alpha,-i\beta) = O(-\alpha,-\beta) \end{array}}}, \\ \langle \nabla_2+\nabla_5+\alpha\nabla_7+\nabla_8 \rangle ^{ { \begin{array}{l} O(\alpha) = O(\eta_5^2\alpha) = O(\eta_5^4\alpha) = \\ O(-\eta_5\alpha) = O(-\eta_5^3\alpha) \end{array}}}, \langle \nabla_3 \rangle, \\ \langle \nabla_3+\nabla_4+\alpha\nabla_5 \rangle ^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \\ \langle \nabla_3+\nabla_4+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_3+\nabla_5 \rangle, \langle \nabla_3+\nabla_6 \rangle, \\ \langle \nabla_3+\nabla_6+\nabla_7 \rangle, \langle \nabla_4+\nabla_5+\alpha\nabla_7+\nabla_8 \rangle^{O(\alpha) = O(-\alpha)}, \langle \nabla_4+\nabla_6+\nabla_7 \rangle, \\ \langle \nabla_4+\nabla_7 \rangle, \langle \nabla_4+\nabla_7+\nabla_8 \rangle, \langle \nabla_4+\nabla_8 \rangle, \langle \nabla_5+\nabla_7+\nabla_8 \rangle, \langle \nabla_5+\nabla_8 \rangle, \\ \langle \nabla_6+\nabla_7 \rangle, \langle \nabla_7 \rangle, \langle \nabla_7+\nabla_8 \rangle, \langle \nabla_8 \rangle, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{246}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{247}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{248} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{249}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{250} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{251}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{252}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{253}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{254} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{255}^{\alpha} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{256}^{\alpha} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{257} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{258} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{259} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{260}^{\alpha} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{261} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{262} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{263} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{264} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{265} & : & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{266} & : & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{267} & : & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{268} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{269} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{270} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{18} & \begin{array}{l} e_1e_1 = e_4 \\ e_1e_2 = e_3 \\ e_3e_3 = e_4 \end{array} & \begin{array}{lcl} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{18})& = &\\ {\langle [\Delta_{11}],[\Delta_{13}],[\Delta_{22}],[\Delta_{23}]\rangle}\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{18})& = &\mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{18})\oplus\\ {\langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle} \end{array} & \phi_{\pm} = \begin{pmatrix} x&0&0&0\\ 0&\pm 1&0&0\\ 0&0&\pm x &0\\ t&s&0&x^2 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{22}], \\ \nabla_5 = [\Delta_{23}], & \nabla_6 = [\Delta_{24}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array} |
Take
\phi_{\pm}^T\begin{pmatrix} \alpha_1&0&\alpha_2&\alpha_3\\ 0&\alpha_4&\alpha_5&\alpha_6\\ \alpha_2&\alpha_5&0&\alpha_7\\ \alpha_3&\alpha_6&\alpha_7&\alpha_8 \end{pmatrix}\phi_{\pm} = \begin{pmatrix} \alpha_1^*&\alpha{*}&\alpha^{*}_2&\alpha_3^*\\ \alpha{*}&\alpha^*_4&\alpha^*_5&\alpha_6^*\\ \alpha^{*}_2&\alpha^*_5&0&\alpha^*_7\\ \alpha^*_3&\alpha^*_6&\alpha^*_7&\alpha^*_8 \end{pmatrix}, |
we have
\begin{array}{ll} \alpha_1^* = \alpha_1x^2+2\alpha_3xt+\alpha_8t^2, & \alpha_2^* = \pm (\alpha_2x+\alpha_7t) x, \\ \alpha_3^* = (\alpha_3x+\alpha_8t)x^2, & \alpha_4^* = \alpha_4\pm2\alpha_6s+\alpha_8s^2, \\ \alpha_5^* = (\alpha_5\pm\alpha_7s)x, & \alpha_6^* = (\pm\alpha_6+\alpha_8s)x^2, \\ \alpha_7^* = \pm\alpha_7x^3, & \alpha_8^* = \alpha_8x^4. \end{array} |
We are interested in
2.
3.
4.
Summarizing all cases, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_7+\nabla_8 \rangle ^{ { \begin{array}{l}O(\alpha,\beta,\gamma,\mu) = O(-\alpha,-\beta,-\gamma,\mu) = \\ O(-\alpha,\beta,\gamma,-\mu) = O(\alpha,-\beta,-\gamma,-\mu) \end{array}}}, \\ \langle \alpha\nabla_1+\beta\nabla_2+\nabla_5+\nabla_6 \rangle^{O(\alpha,\beta) = O(-\alpha,\beta)}, \langle \alpha\nabla_1+\beta\nabla_2+\nabla_6 \rangle^{O(\alpha,\beta) = O(-\alpha,\beta)}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_6+\nabla_7 \rangle^{O(\alpha,\beta,\gamma) = O(-\alpha,\beta,-\gamma)}, \langle \nabla_2+\nabla_3+\alpha\nabla_4+\beta\nabla_5 \rangle^{O(\alpha,\beta) = O(-\alpha,\beta)}, \langle \alpha\nabla_2+\nabla_3 +\beta\nabla_5+\nabla_6 \rangle, \\ \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_7+\nabla_8 \rangle^{O(\alpha,\beta,\gamma) = O(\alpha,i\beta, i\gamma) = O(\alpha,-i\beta, -i\gamma)}, \langle \nabla_3 \rangle, \\ \langle \nabla_3+\nabla_4+\alpha\nabla_5 \rangle ^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \\ \langle \alpha\nabla_3+\nabla_4+\beta\nabla_6+\nabla_7 \rangle^{{ \begin{array}{l} O(\alpha,\beta) = O(-\alpha,-\beta) = O(-\alpha,\eta_3\beta) = \\O(-\alpha,-\eta_3^2\beta) = O(\alpha,-\eta_3\beta) = O(\alpha,\eta_3^2\beta)\end{array}}}, \langle \nabla_3+\nabla_5 \rangle, \\ \langle \alpha\nabla_3+\nabla_6+\nabla_7 \rangle ^{O(\alpha,\beta) = O(-\alpha,\beta)}, \langle \alpha\nabla_3+\nabla_7 \rangle^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_4+\alpha\nabla_5+\beta\nabla_7+\nabla_8 \rangle ^{ { \begin{array}{l}O(\alpha,\beta) = O(i\alpha,-i\beta) = O(-i\alpha,i\beta) = O(-\alpha,-\beta) = \\ O(\alpha,-\beta) = O(i\alpha,i\beta) = O(-i\alpha,-i\beta) = O(-\alpha,\beta) \end{array}}}, \\ \langle \nabla_5+\alpha\nabla_7+\nabla_8 \rangle^{ O(\alpha) = O(\eta_3\alpha) = O(-\eta_3^2\alpha) = O(-\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_7+\nabla_8 \rangle, \langle \nabla_8\rangle, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{271}^{\alpha, \beta, \gamma,\mu} & : & e_1e_1 = e_4+e_5 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_2e_2 = \beta e_5 \\ && e_2e_3 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{272}^{\alpha, \beta} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{273}^{\alpha, \beta} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{274}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_4+e_5 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = \beta e_5 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{275}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{276}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{277}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{278} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{279}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{280}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{281} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{282}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{283}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{284}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{285}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{286} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{287} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{22}], \\ \nabla_5 = [\Delta_{23}], & \nabla_6 = [\Delta_{24}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} \alpha_1&0&\alpha_2&\alpha_3\\ 0&\alpha_4&\alpha_5&\alpha_6\\ \alpha_2&\alpha_5&0&\alpha_7\\ \alpha_3&\alpha_6&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{*}_2&\alpha_3^*\\ \alpha^{*}&\alpha^*_4&\alpha^*_5&\alpha_6^*\\ \alpha^{*}_2&\alpha^*_5&0&\alpha^*_7\\ \alpha^*_3&\alpha^*_6&\alpha^*_7&\alpha^*_8 \end{pmatrix}, |
then, in the case
\begin{array}{llll} \alpha_1^* = \alpha_1+2\alpha_3t+\alpha_8t^2, & \alpha_2^* = \alpha_2+\alpha_7t, & \alpha_3^* = \alpha_3+\alpha_8t, & \alpha_4^* = \alpha_4+2\alpha_6s+\alpha_8s^2, \\ \alpha_5^* = \alpha_5+\alpha_7s, & \alpha_6^* = \alpha_6+\alpha_8s, & \alpha_7^* = \alpha_7, & \alpha_8^* = \alpha_8. \end{array} |
For define the main families of representatives, we will use
(\alpha_3,\alpha_6,\alpha_7,\alpha_8)\neq(0,0,0,0) . |
Let us consider the following cases:
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_7+\nabla_8 \rangle ^{{ \begin{array}{l} O(\alpha,\beta,\gamma,\mu,\nu) = O(\alpha,-\beta,\gamma,\mu,-\nu) = \\ O(\alpha,\beta,\gamma,-\mu,-\nu) = O(\alpha,-\beta,\gamma,-\mu,\nu) = \\ O(\gamma,\mu,\alpha,\beta,\nu) = O(\gamma,-\mu,\alpha,\beta,-\nu) = \\ O(\gamma,\mu,\alpha,-\beta,-\nu) = O(\gamma,-\mu,\alpha,-\beta,\nu) \end{array}}}, \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_6+\nabla_7 \rangle ^{{ \begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(-\alpha,-\beta,-\gamma,\mu) = \\ O(-\alpha,\beta,-\gamma,-\mu) = O(\alpha,-\beta,\gamma,-\mu) = \\ O(\gamma,\mu,\alpha,\beta) = O(-\gamma,-\mu,-\alpha,\beta) = \\ O(-\gamma,\mu,-\alpha,-\beta) = O(\gamma,-\mu,\alpha,-\beta) \end{array}}}, \\ \langle \alpha\nabla_2+\nabla_3+\beta\nabla_4+\gamma\nabla_5 \rangle^{{ \begin{array}{l} O(\alpha,\beta,\gamma) = O(-\alpha,\beta,\gamma) = O(-\alpha,-\beta,\gamma) = O(\alpha,-\beta,\gamma) = \end{array}}}, \\ \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_5+\nabla_6 \rangle_{\beta \neq0}^{{ \begin{array}{l} O(\alpha,\beta,\gamma) = O(\alpha,-\beta,-\gamma) = O(\frac{\gamma}{\beta},\frac{1}{\beta},\frac{\alpha}{\beta}) = O(\frac{\gamma}{\beta},-\frac{1}{\beta},-\frac{\alpha}{\beta}) \end{array}}},\end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{288}^{\alpha, \beta, \gamma,\mu,\nu} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 & e_2e_2 = e_4+\gamma e_5 \\ & & e_2e_3 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{289}^{\alpha, \beta, \gamma,\mu} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_4 = \beta e_5 & e_2e_2 = e_4+\gamma e_5 \\ & & e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{290}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{291}^{\alpha, \beta\neq0 ,\gamma} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ & & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{01} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_3 \\ e_2e_3 = e_4 \end{array} &\begin{array}{l}\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{01}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(1,2),(2,3)\} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0 & x^2&0&0\\ z&0 & x^3&0\\ t&0 & x^2z & x^5 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&\alpha_3&0&\alpha_4\\ \alpha_1&0&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1&\alpha^*_2\\ \alpha^{**}&\alpha^*_3&\alpha^{***}&\alpha^*_4\\ \alpha^{*}_1&\alpha^{***}&\alpha^*_5&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{lll} { \alpha_1^* = \big((\alpha_1x+\alpha_5z+\alpha_6t)x+(\alpha_2x+\alpha_6z+\alpha_7t)z\big)x^2, }\\ \alpha_2^* = (\alpha_2x+\alpha_6z+\alpha_7t)x^5, & \alpha_3^* = \alpha_3x^4, & \alpha_4^* = \alpha_4x^7, \\ \alpha_5^* = (\alpha_5x^{2}+2\alpha_6xz+\alpha_7z^2)x^4, & \alpha_6^* = (\alpha_6x+\alpha_7z)x^{7}, & \alpha_7^* = \alpha_7x^{10}. \end{array} |
We are interested in
(b) if
2.
(b) if
(c) if
3.
4.
Summarizing all cases, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+ \alpha \nabla_2 + \beta \nabla_3 + \nabla_4 -\alpha \nabla_5 \rangle^ {O(\alpha, \beta) = O(-\eta_3\alpha, \eta_3 \beta) = O(\eta_3^2\alpha,-\eta_3^2\beta)}, \\ \langle \nabla_1+ \nabla_2 + \alpha \nabla_3 - \nabla_5 \rangle, \\ \langle \nabla_1+ \alpha \nabla_3 + \beta \nabla_4 + \gamma \nabla_5 + \nabla_7 \rangle^ {{ \begin{array}{l} O(\alpha, \beta, \gamma) = O(\alpha, \beta, -\eta_3\gamma) = O(\alpha, -\beta, -\eta_3\gamma) = \\ O(\alpha, -\beta, \eta_3^2 \gamma) = O(\alpha, \beta, \eta_3^2\gamma) = O(\alpha, -\beta, \gamma) \end{array} }}, \\ \langle \alpha \nabla_2 + \nabla_3 + \nabla_4 +\beta \nabla_5 \rangle ^{O(\alpha, \beta) = O(-\eta_3\alpha, -\eta_3 \beta) = O(\eta_3^2\alpha, \eta_3^2\beta)}, \langle \nabla_2+ \nabla_3 + \alpha \nabla_5 \rangle, \\ \langle \nabla_2+ \nabla_4 + \alpha \nabla_5 \rangle, \langle \nabla_2 + \alpha \nabla_5 \rangle, \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle ^{{ \begin{array}{l} O(\alpha, \beta) = O(-i\alpha, -\beta) = \\ O(i\alpha, -\beta) = O(-\alpha, \beta) \end{array}}}, \\ \langle \nabla_3 + \alpha\nabla_4 + \beta \nabla_5 + \nabla_7 \rangle^ {{ \begin{array}{l} O(\alpha, \beta) = O(\alpha, -\eta_3 \beta) = O(-\alpha, -\eta_3\beta) = \\ O(-\alpha, \eta_3^2 \beta) = O(\alpha, \eta_3^2 \beta) = O(-\alpha, \beta) \end{array}}}, \langle \nabla_4 \rangle, \langle \nabla_4 + \nabla_5 \rangle, \\ \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle, \langle \nabla_4 + \alpha \nabla_5 + \nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_5+\nabla_6 \rangle, \langle \nabla_5 + \nabla_7 \rangle, \\ \langle \nabla_6 \rangle, \langle \nabla_7 \rangle. \end{array} |
Hence, we have the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{292}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = \beta e_5 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = -\alpha e_5 \\ {\mathbf{N}}_{293}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = e_4 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{294}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{295}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{296}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{297}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{298}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{299}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{300}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_4 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{301} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{302} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{303}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{304}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{305} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{306} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{307} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{308} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{02} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_3 \\ e_1e_3 = e_4 \\ e_2e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{02}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(1,2),(1,3)\} \end{array} & \phi = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ z&0&1&0\\ t&2z&z&1 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{14}], & \nabla_2 = [\Delta_{22}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&0&0&\alpha_1\\ 0&\alpha_2&\alpha_3&\alpha_4\\ 0&\alpha_3&\alpha_5&\alpha_6\\ \alpha_1&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{***}&\alpha^*_1\\ \alpha^{**}&\alpha^*_2&\alpha^*_3+\alpha^{***}&\alpha^*_4\\ \alpha^{***}&\alpha^*_3+\alpha^{***}&\alpha^*_5&\alpha^*_6\\ \alpha^*_1&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{lcl} \alpha_1^* & = & \alpha_1+\alpha_6z+\alpha_7t, \\ \alpha_2^* & = & \alpha_2+4\alpha_4z+4\alpha_7z^2, \\ \alpha_3^* & = & \alpha_3+2\alpha_6z+(\alpha_4+2\alpha_7z)z-(\alpha_5z+\alpha_6t)-(\alpha_1+\alpha_6z+\alpha_7t)z, \\ \alpha_4^* & = & \alpha_4+2\alpha_7z, \\ \alpha_5^* & = & \alpha_5+2\alpha_6z+\alpha_7z^2, \\ \alpha_6^* & = & \alpha_6+\alpha_7z, \\ \alpha_7^* & = & \alpha_7. \end{array} |
We are interested in
\langle \nabla_1+\alpha\nabla_2+\beta\nabla_3-\nabla_5 \rangle; |
2. if
3. if
z = -{\alpha_1}{\alpha_6}^{-1}, t = ({\alpha_3\alpha_6-\alpha_1(2\alpha_6+\alpha_4-\alpha_5)}){\alpha_6^{-1}}, |
we have the family of representatives
4. if
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+ \alpha \nabla_2 + \beta \nabla_3 - \nabla_5 \rangle, \langle \nabla_1+ \alpha \nabla_2 + \beta \nabla_5 \rangle_{\beta\neq -1}, \langle \alpha \nabla_1+ \beta \nabla_3 + \nabla_4 + \gamma\nabla_5 \rangle, \\ \langle \alpha \nabla_2 + \beta \nabla_3 + \gamma\nabla_4 + \mu \nabla_5 + \nabla_7\rangle, \langle \alpha \nabla_2 + \beta \nabla_4 +\gamma\nabla_5+ \nabla_6\rangle, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{309}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = e_4+\beta e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{310}^{\alpha, \beta\neq-1} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = e_4 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{311}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{312}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4+\beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{313}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{03} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_3 \\ e_3e_3 = e_4 \end{array} &\begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{03}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ {(i,j) \notin \{ (1,1),(1,2),(3,3)}\} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&0&x^3&0\\ t&0&0&x^6 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ \alpha_1&\alpha_4&0&\alpha_6\\ \alpha_2&\alpha_5&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1&\alpha^*_2\\ \alpha^{**}&\alpha^*_3&\alpha^{*}_4&\alpha^*_5\\ \alpha^{*}_1&\alpha^{*}_4&0&\alpha^*_6\\ \alpha^*_2&\alpha^*_5&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{llll} \alpha_1^* = (\alpha_1x+\alpha_6t)x^3, & \alpha_2^* = (\alpha_2x+\alpha_7t)x^6, & \alpha_3^* = \alpha_3x^4, & \alpha_4^* = \alpha_4x^5, \\ \alpha_5^* = \alpha_5x^8, & \alpha_6^* = \alpha_6x^{9}, & \alpha_7^* = \alpha_7x^{12}. \end{array} |
We are interested in
2.
3.
4.
x = \sqrt[8]{{(\alpha_1\alpha_7-\alpha_2\alpha_6)}{\alpha^{-2}_7}}, t = -{\alpha_2\sqrt[8]{(\alpha_1\alpha_7-\alpha_2\alpha_6)\alpha_7^{-10}}}, |
we have the family of representatives
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+\nabla_2 + \alpha \nabla_3 + \beta \nabla_4 \rangle ^{O(\alpha, \beta) = O(\alpha, -\eta_3 \beta) = O(\alpha, \eta_3^2\beta)}, \\ \langle \nabla_1+ \alpha \nabla_2 + \beta \nabla_3 + \gamma \nabla_4 + \nabla_5\rangle ^{O(\alpha, \beta, \gamma) = O(-i\alpha, \beta, i\gamma) = O(i\alpha, \beta,-i \gamma) = O(-\alpha, \beta, -\gamma)}, \\ \langle \nabla_1+\alpha \nabla_3+ \beta \nabla_4 + \gamma\nabla_5 + \mu\nabla_6 + \nabla_7\rangle^{ {\begin{array}{l} O(\alpha,\beta,\gamma, \mu) = O(\alpha,\eta_4^3\beta,-\gamma, -\eta_4^3\mu) = \\ O(\alpha,-\eta_4^3\beta,-\gamma, \eta_4^3\mu) = O(\alpha,\eta_4 \beta,-\gamma, -\eta_4\mu) = \\ O(\alpha,-\eta_4\beta,-\gamma, \eta_4\mu) = O(\alpha,i\beta,\gamma, i\mu) = \\ O(\alpha,-i\beta,\gamma, -i\mu) = O(\alpha,-\beta, \gamma, -\mu) \end{array}} }, \\ \langle \nabla_2\rangle, \langle \nabla_2+\nabla_3 +\alpha\nabla_4\rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_2+\alpha\nabla_3 +\beta\nabla_4 + \nabla_5\rangle, \\ \langle \nabla_2+\alpha \nabla_3+ \beta \nabla_4 + \gamma\nabla_5 + \nabla_6\rangle^{O(\alpha,\beta,\gamma) = O(-\alpha,\beta,-\gamma)}, \langle \nabla_2+ \nabla_4\rangle, \\ \langle \nabla_3+\alpha \nabla_4+\nabla_5\rangle^{ { \begin{array}{l} O(\alpha) = O(-\alpha) = \\ O(i\alpha) = O(-i\alpha) \end{array}}}, \\ \langle \nabla_3+\alpha \nabla_4+ \beta \nabla_5+ \nabla_6\rangle ^{{ \begin{array}{l} O(\alpha, \beta) = O(\eta_5^4\alpha, -\eta_5\beta) = O(-\eta_5^3\alpha, \eta_5^2\beta) = \\ O(\eta_5^2\alpha, -\eta_5^3\beta) = O(-\eta_5\alpha, \eta_5^4\beta) \end{array}}}, \\ \langle \nabla_3+\alpha \nabla_4+ \beta \nabla_5 +\gamma\nabla_6 + \nabla_7\rangle^{ {\begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_4^3\alpha,-\beta,-\eta_4^3\gamma) = O(-\eta_4^3\alpha,-\beta,\eta_4^3\gamma) = \\ O(\eta_4\alpha,-\beta,-\eta_4\gamma) = O(-\eta_4\alpha,-\beta,\eta_4\gamma) = \\ O(i\alpha,\beta,i\gamma) = O(-i\alpha,\beta,-i\gamma) = O(-\alpha,\beta,-\gamma) \end{array}} }, \\ \langle \nabla_4+ \nabla_5\rangle, \langle \nabla_4+\alpha \nabla_5+ \nabla_6\rangle^{O(\alpha) = O(i\alpha) = O(-\alpha) = O(-i\alpha)}, \\ \langle \nabla_4+\alpha \nabla_5+ \beta\nabla_6 + \nabla_7\rangle^{ {\begin{array}{l} O(\alpha,\beta) = O(\eta^4_7\alpha,-\eta^3_7\beta) = O(-\eta_7\alpha,\eta_7^6\beta) = O(-\eta_7^5\alpha,\eta^2_7\beta) = \\ O(\eta^2_7\alpha,-\eta^5_7\beta) = O(\eta_7^6\alpha,-\eta_7\beta) = O(-\eta^3_7\alpha,\eta^4_7\beta) \end{array}}}, \langle \nabla_5\rangle, \\ \langle \nabla_5 + \nabla_6\rangle, \langle \nabla_5+\alpha \nabla_6+ \nabla_7\rangle^{O(\alpha) = O(i\alpha) = O(-\alpha) = O(-i\alpha)}, \langle \nabla_6\rangle, \langle \nabla_6 +\nabla_7\rangle, \langle \nabla_7\rangle, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{314}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{315}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = \beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{316}^{\alpha, \beta, \gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{317} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{318}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{319}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{320}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{321} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{322}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{323}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{324}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{325} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{326}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{327}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{328} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{329} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}^\alpha_{330} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{331} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{332} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{333} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{04} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_3 \\ e_2e_2 = e_4 \\ e_3e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{04}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{(1,1),(1,2),(3,3)\} \end{array} & \phi_{\pm} = \begin{pmatrix} \pm1&0&0&0\\ 0&1&0&0\\ 0&0&\pm1&0\\ t&0&0&1 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ \alpha_1&\alpha_4&0&\alpha_6\\ \alpha_2&\alpha_5&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1&\alpha^*_2\\ \alpha^{**}&\alpha^*_3&\alpha^{*}_4&\alpha^*_5\\ \alpha^{*}_1&\alpha^{*}_4&0&\alpha^*_6\\ \alpha^*_2&\alpha^*_5&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{llll} \alpha_1^* = \alpha_1\pm\alpha_6t, & \alpha_2^* = \pm\alpha_2+\alpha_7t, & \alpha_3^* = \alpha_3, & \alpha_4^* = \pm\alpha_4, \\ \alpha_5^* = \alpha_5, & \alpha_6^* = \pm \alpha_6, & \alpha_7^* = \alpha_7. \end{array} |
We are interested in
\langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_4 \rangle; |
\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_3+\mu\nabla_4+\nabla_5 \rangle; |
\langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle; |
\langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle. |
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_4 \rangle^{O(\alpha, \beta, \gamma) = O(-\alpha, -\beta, \gamma)}, \\\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_3+\mu\nabla_4+\nabla_5 \rangle ^{O(\alpha, \beta, \gamma,\mu) = O(\alpha, -\beta, \gamma, -\mu)} \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle^{O(\alpha, \beta, \gamma,\mu,\nu) = O(\alpha, \beta, -\gamma,\mu,-\nu)}, \\ \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle ^{O(\alpha, \beta, \gamma, \mu) = O(\alpha, -\beta, \gamma, -\mu)}, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{334}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{335}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ && e_2e_2 = e_4+\gamma e_5 & e_2e_3 = \mu e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{336}^{\alpha, \beta,\gamma, \mu, \nu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 \\& & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = \mu e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{337}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4+\beta e_5 \\ && e_2e_3 = \gamma e_5 & e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms}\\ \hline {\mathbf{N}}^{4}_{05} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_3 = e_4 \\ e_2e_2 = e_3 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{05}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ {(i,j) \notin \{ (1,1),(1,3),(2,2)} \} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ z&0&x^4&0\\ t&2xz&0&x^5 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{12}] & \nabla_2 = [\Delta_{14}] & \nabla_3 = [\Delta_{23}] & \nabla_4 = [\Delta_{24}] \\ \nabla_5 = [\Delta_{33}] & \nabla_6 = [\Delta_{34}] & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&\alpha_1&0&\alpha_2\\ \alpha_1&0&\alpha_3&\alpha_4\\ 0&\alpha_3&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^*_1&\alpha^{***}&\alpha^*_2\\ \alpha^*_1&\alpha^{**}&\alpha^*_3&\alpha^*_4\\ \alpha^{***}&\alpha^*_3&\alpha^*_5&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{lll} { \alpha_1^* = (\alpha_1x+\alpha_3z+\alpha_4t)x^2+2(\alpha_2x+\alpha_6z+\alpha_7t)xz, }\\ \alpha_2^* = (\alpha_2x+\alpha_6z+\alpha_7t)x^5, & \alpha_3^* = (\alpha_3x+2\alpha_6z)x^5, & \alpha_4^* = (\alpha_4x+2\alpha_7z)x^6, \\ \alpha_5^* = \alpha_5x^8, & \alpha_6^* = \alpha_6x^9, & \alpha_7^* = \alpha_7x^{10}. \end{array} |
We are interested in
(b)
2.
3.
(b)
4.
x = \frac{\alpha_6}{\alpha_7}, z = -\frac{\alpha_4\alpha_6}{2\alpha_7^2}, t = \frac{\alpha_6(\alpha_4\alpha_6-2\alpha_2\alpha_7)}{2\alpha^3_7}, |
we have the family of representatives
x = \sqrt[4]{\frac{\alpha_3\alpha_7-\alpha_4\alpha_7}{\alpha^2_7}}, z = -\frac{\alpha_4\sqrt[4]{\alpha_3\alpha_7-\alpha_4\alpha_7}}{2\alpha_7\sqrt[4]{2\alpha_7^2}}, t = \frac{(\alpha_4\alpha_6-2\alpha_2\alpha_7)\sqrt[4]{\alpha_3\alpha_7-\alpha_4\alpha_6}}{2\alpha^2_7\sqrt[4]{2\alpha_7^2}}, |
we have the family of representatives
\begin{array}{c} x = \sqrt[7]{\frac{2\alpha_1\alpha_7^2-\alpha_3\alpha_4\alpha_7+\alpha_4^2\alpha_6-2\alpha_2\alpha_4\alpha_7}{2\alpha^3_7}}, \\ z = -\frac{\alpha_4\sqrt[7]{2\alpha_1\alpha_7^2-\alpha_3\alpha_4\alpha_7+\alpha_4^2\alpha_6-2\alpha_2\alpha_4\alpha_7}}{2\alpha_7\sqrt[7]{2\alpha_7^3}}, \\ t = \frac{(\alpha_4\alpha_6-2\alpha_2\alpha_7)\sqrt[7]{2\alpha_1\alpha_7^2-\alpha_3\alpha_4\alpha_7+\alpha_4^2\alpha_6-2\alpha_2\alpha_4\alpha_7}}{2\alpha^3_7\sqrt[7]{2\alpha_7^3}}, \end{array} |
we have the family of representatives
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+\nabla_2-2\nabla_3+\alpha\nabla_5 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta^2_3\alpha)}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_5+\nabla_6 \rangle^{{\begin{array}{l} O(\alpha,\beta) = O(\alpha,-\eta_3\beta) = O(-\alpha,\eta_3\beta) = \\ O(-\alpha,-\eta_3^2\beta) = O(\alpha,\eta_3^2\beta) = O(-\alpha,-\beta) \end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_7^4\alpha,\eta^2_7\beta,-\eta_7\gamma) = O(-\eta_7\alpha,\eta^4_7\beta,\eta^2_7\gamma) = \\ O(-\eta_7^5\alpha,\eta^6_{7}\beta,-\eta^3_7\gamma) = O(\eta^2_7\alpha,-\eta_7\beta,\eta^4_7\gamma) = \\ O(\eta_7^6\alpha,-\eta^3_7\beta,-\eta^5_7\gamma) = O(-\eta^3_7\alpha,-\eta^5_7\beta,\eta^6_7\gamma) \end{array}}}, \\ \langle \nabla_2+\alpha\nabla_3 \rangle, \langle \nabla_2+\alpha\nabla_3+ \nabla_4+\beta\nabla_5 \rangle, \langle\nabla_2+\alpha\nabla_3+\nabla_5 \rangle, \langle \nabla_3+ \nabla_4+\alpha\nabla_5 \rangle, \\ \langle \alpha\nabla_3+\nabla_4+\beta\nabla_5+\nabla_6 \rangle^{O(\alpha,\beta) = O(-\alpha,-\beta)}, \langle \nabla_3+\alpha\nabla_5+\nabla_6 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta^2_3\alpha)}, \\ \langle \nabla_3+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle^{O(\alpha,\beta) = O(-\alpha,-i\beta) = O(-\alpha,i \beta) = O(\alpha,-\beta)}, \langle \nabla_4 \rangle, \langle \nabla_4+\nabla_5 \rangle, \\ \langle \nabla_5+\nabla_6 \rangle, \langle \alpha\nabla_5+\nabla_6+\nabla_7 \rangle, \langle \nabla_5+\nabla_7 \rangle, \langle\nabla_6 \rangle, \langle\nabla_7 \rangle,\end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{338}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = -2e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{339}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{340}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{341}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{342}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{343}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{344}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{345}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{346}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{347}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{348} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ {\mathbf{N}}_{349} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{350} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{351}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{352} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{353} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_4 = e_5 \\ {\mathbf{N}}_{354} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{06} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_4 \\ e_1e_3 = e_4 \\ e_2e_2 = e_3 \end{array} &\begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{06}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{(1,1),(1,2),(2,2)\} \end{array} & \phi_{\pm} = \begin{pmatrix} \pm1&0&0&0\\ 0&1&0&0\\ z&0&1&0\\ t&\pm2z&0&\pm1 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi_{\pm}^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&0&\alpha_3&\alpha_4\\ \alpha_1&\alpha_3&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi_{\pm} = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1+\alpha^{**}&\alpha^*_2\\ \alpha^{**}&\alpha^{***}&\alpha^*_3&\alpha^*_4\\ \alpha^{*}_1+\alpha^{**}&\alpha^*_3&\alpha^*_5&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{lll} { \alpha_1^* = \pm \alpha_1- \alpha_3z- \alpha_4t+ \alpha_5z+ \alpha_6t-2 ( \alpha_2\pm \alpha_6z\pm \alpha_7t)z, }\\ \alpha_2^* = \alpha_2\pm\alpha_6z \pm\alpha_7t, & \alpha_3^* = \alpha_3 \pm 2\alpha_6z, & \alpha_4^* = 2\alpha_7z \pm \alpha_4, \\ \alpha_5^* = \alpha_5, & \alpha_6^* = \pm \alpha_6, & \alpha_7^* = \alpha_7. \end{array} |
Since
\langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+(\beta+2)\nabla_5 \rangle; |
2. if
3. if
4. if
\langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle. |
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+(\beta+2)\nabla_5 \rangle_{\alpha \neq 0}^{O(\alpha, \beta) = O(-\alpha, \beta)}, \\ \langle \alpha\nabla_1+\beta\nabla_3+\nabla_4+\gamma\nabla_5+\nabla_6 \rangle^{O(\alpha, \beta, \gamma) = O(\alpha, -\beta,-\gamma)}_{\alpha\neq 0}, \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle^{O(\alpha, \beta, \gamma, \mu) = O(-\alpha, \beta, \gamma, -\mu)}, \langle \nabla_2+\alpha\nabla_3+\beta\nabla_5 \rangle, \\ \langle \alpha\nabla_2+\beta\nabla_3+\nabla_4+\gamma\nabla_5 \rangle^{O(\alpha, \beta, \gamma) = O(-\alpha, -\beta,- \gamma)}, \\ \langle \alpha \nabla_3+\beta \nabla_4+\gamma\nabla_5+\nabla_6 \rangle^{O(\alpha, \beta, \gamma) = O(-\alpha, \beta, -\gamma)}, \end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{355}^{\alpha\neq0, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4+\alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & { e_3e_3 = (\beta+2)e_5 } \\ {\mathbf{N}}_{356}^{\alpha\neq0, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4+\alpha e_5 & e_2e_2 = e_3 & e_2e_3 = \beta e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{357}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_3 = e_4+\alpha e_5 & e_2e_2 = e_3 & e_2e_3 = \beta e_5 \\ & & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 & \\ {\mathbf{N}}_{358}^{\alpha,\beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{359}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 & \\ {\mathbf{N}}_{360}^{\alpha,\beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 & \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{07} & \begin{array}{l} e_1e_1 = e_2 \\ e_2e_2 = e_3 \\ e_2e_3 = e_4 \end{array} &\begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{07}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(2,2),(2,3) \} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&0&x^4&0\\ t&0&0&x^6 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&\alpha_1&\alpha_2&\alpha_3\\ \alpha_1&0&0&\alpha_4\\ \alpha_2&0&\alpha_5&\alpha_6\\ \alpha_3&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^*_1&\alpha^{*}_2&\alpha^*_3\\ \alpha^*_1&0&0&\alpha^*_4\\ \alpha^{*}_2&0&\alpha^*_5&\alpha^*_6\\ \alpha^*_3&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{llll} \alpha_1^* = (\alpha_1x+\alpha_4t)x^2, & \alpha_2^* = (\alpha_2x+\alpha_6t)x^4, & \alpha_3^* = (\alpha_3x+\alpha_7t)x^6, & \alpha_4^* = \alpha_4x^8, \\ \alpha_5^* = \alpha_5x^8, & \alpha_6^* = \alpha_6x^{10}, & \alpha_7^* = \alpha_7x^{12}. \end{array} |
We are interested in
2.
3.
4.
\langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle; |
x = \sqrt[9]{(\alpha_1\alpha_7-\alpha_3\alpha_4)\alpha^{-2}_7}, t = -\alpha_3\sqrt[9]{(\alpha_1\alpha_7-\alpha_3\alpha_4)\alpha_7^{-11} }, |
we have family of representatives
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\nabla_3+\beta\nabla_5 \rangle ^{O(\alpha,\beta) = O(-\alpha,i\beta) = O(-\alpha,-i\beta) = O(\alpha,-\beta)}, \\ \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle ^{{\begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(-\eta_9^7\alpha,\eta^4_9\beta,\eta^4_9\gamma,\eta^2_9\mu) = \\ O(-\eta^5_9\alpha,\eta_9^8\beta,\eta_9^8\gamma,\eta^4_9\mu) = O(-\eta_3\alpha,-\eta_3\beta,-\eta_3\gamma,\eta^2_3\mu) = \\ O(-\eta_9\alpha,-\eta^7_9\beta,-\eta^7_9\gamma,\eta^8_9\mu) = O(\eta_9^8\alpha,\eta^2_9\beta,\eta^2_9\gamma,-\eta_9\mu) = \\ O(\eta^2_3\alpha,\eta^2_3\beta,\eta^2_3\gamma,-\eta_3\mu) = O(\eta^4_9\alpha,-\eta_9\beta,-\eta_9\gamma,-\eta^5_9\mu) = \\ O(\eta^2_9\alpha,-\eta^5_9\beta,-\eta^5_9\gamma,-\eta^7_9\mu) \end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle ^{{\begin{array}{l} O(\alpha,\beta,\gamma) = O(-\eta_7^3\alpha,\eta^2_7\beta,\eta^2_7\gamma) = \\ O(\eta_7^6\alpha,\eta^4_7\beta,\eta^4_7\gamma) = O(\eta^2_7\alpha,\eta^6_{7}\beta,\eta^6_7\gamma) = \\ O(-\eta^5_7\alpha,-\eta_7\beta,-\eta_7\gamma) = O(-\eta_7\alpha,-\eta^3_7\beta,-\eta^3_7\gamma) = \\ O(\eta^4_7\alpha,-\eta^5_7\beta,-\eta^5_7\gamma) \end{array}}}, \\ \langle \nabla_2+\alpha\nabla_3+ \nabla_4+\beta\nabla_5 \rangle^{O(\alpha,\beta) = O(-\eta_3\alpha,\beta) = O(\eta^2_3\alpha,\beta)}, \langle \nabla_2+\nabla_3+\alpha\nabla_5 \rangle^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_7^4\alpha,\eta^4_7\beta,\eta^2_7\gamma) = O(-\eta_7\alpha,-\eta_7\beta,\eta^4_7\gamma) = \\ O(-\eta^5_7\alpha,-\eta^5_{7}\beta,\eta^6_7\gamma) = O(\eta^2_7\alpha,\eta^2_7\beta,-\eta_7\gamma) = \\ O(\eta_7^6\alpha,\eta_7^6\beta,-\eta^3_7\gamma) = O(-\eta^3_7\alpha,-\eta^3_7\beta,-\eta^5_7\gamma) \end{array}}}, \langle \nabla_3 \rangle, \\ \langle \nabla_3+ \nabla_4+\alpha\nabla_5 \rangle, \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle^{O(\alpha,\beta) = O(-\eta_3\alpha,-\eta_3\beta) = O(\eta^2_3\alpha,\eta^2_3\beta)}, \\ \langle \nabla_3+\nabla_5 \rangle, \langle \nabla_4+\alpha\nabla_5 \rangle, \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle, \langle \nabla_4+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle^{O(\alpha,\beta) = O(\alpha,-\beta)}, \\ \langle \nabla_5+\nabla_6 \rangle, \langle \nabla_5+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\alpha) }, \langle \nabla_6 \rangle, \langle \nabla_6 +\nabla_7 \rangle, \langle \nabla_7 \rangle,\end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{361}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{362}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{363}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{364}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{365}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{366}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_1e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{367} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ {\mathbf{N}}_{368}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{369}^{\alpha,\beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{370} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{371}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{372}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{373}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ & & e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{374} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{375}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_3e_3 = e_5 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{376} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{377} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{378} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms}\\ \hline {\mathbf{N}}^{4}_{08} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_3 = e_4 \\ e_2e_2 = e_3 \\ e_2e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{08}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ {(i,j) \notin \{(1,1),(1,3),(2,2)}\} \end{array} & \phi = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ t&0&0&1 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&\alpha_1&0&\alpha_2\\ \alpha_1&0&\alpha_3&\alpha_4\\ 0&\alpha_3&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{*}_1&\alpha^{**}&\alpha^*_2\\ \alpha^{*}_1&0&\alpha^*_3+\alpha^{**}&\alpha^*_4\\ \alpha^{**}&\alpha^*_3+\alpha^{**}&\alpha^*_5&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{llll} \alpha_1^* = \alpha_1+\alpha_4t, & \alpha_2^* = \alpha_2+\alpha_7t, & \alpha_3^* = \alpha_3-\alpha_6t, & \alpha_4^* = \alpha_4, \\ \alpha_5^* = \alpha_5, & \alpha_6^* = \alpha_6, & \alpha_7^* = \alpha_7. \end{array} |
Since
\langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_5 \rangle; |
\langle \alpha\nabla_2+\beta\nabla_3+\nabla_4+\gamma\nabla_5 \rangle; |
\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle; |
\langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle. |
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_5 \rangle,\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle, \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle,\langle \alpha\nabla_2+\beta\nabla_3+\nabla_4+\gamma\nabla_5 \rangle,\end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{379}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4+\beta e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{380}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 \\ & & e_1e_4 = \beta e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{381}^{\alpha, \beta,\gamma,\mu, \nu} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = e_4+\beta e_5 & e_2e_4 = \gamma e_5 \\ && e_3e_3 = \mu e_5 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{382}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{09} & \begin{array}{l} e_1e_1 = e_2 \\ e_2e_2 = e_3 \\ e_3e_3 = e_4 \end{array} &\begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{09}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin\{ (1,1),(2,2),(3,3)\} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&0&x^4&0\\ t&0&0&x^8 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&\alpha_1&\alpha_2&\alpha_3\\ \alpha_1&0&\alpha_4&\alpha_5\\ \alpha_2&\alpha_4&0&\alpha_6\\ \alpha_3&\alpha_5&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^*_1&\alpha^{*}_2&\alpha^*_3\\ \alpha^*_1&0&\alpha^*_4&\alpha^*_5\\ \alpha^{*}_2&\alpha^*_4&0&\alpha^*_6\\ \alpha^*_3&\alpha^*_5&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{llll} \alpha_1^* = (\alpha_1x+\alpha_5t)x^2, & \alpha_2^* = (\alpha_2x+\alpha_6t)x^4, & \alpha_3^* = (\alpha_3x+\alpha_7t)x^8, & \alpha_4^* = \alpha_4x^6, \\ \alpha_5^* = \alpha_5x^{10}, & \alpha_6^* = \alpha_6x^{12}, & \alpha_7^* = \alpha_7x^{16}. \end{array} |
Since
2.
3.
x = \sqrt[9]{(\alpha_1\alpha_6-\alpha_2\alpha_5)\alpha^{-2}_6}, t = -{\alpha_2\sqrt[9]{(\alpha_1\alpha_6-\alpha_2\alpha_5)\alpha_6^{-11}}}, |
we have the family of representatives
\langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle. |
4.
x = \sqrt[10]{{\alpha_4}{\alpha_7^{-1}}}, t = -{\alpha_3\sqrt[10]{\alpha_4 \alpha_7^{-11}}}, |
we have the family of representatives
x = \sqrt[11]{(\alpha_2\alpha_7-\alpha_3\alpha_6)\alpha^{-2}_7 }, t = -{\alpha_3\sqrt[11]{(\alpha_2\alpha_7-\alpha_3\alpha_6) \alpha_7^{-13} }}, |
we have the family of representatives
\langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle; |
x = \sqrt[13]{({\alpha_1\alpha_7-\alpha_3\alpha_5}){\alpha^{-2}_7}}, t = -{\alpha_3\sqrt[13]{(\alpha_1\alpha_7-\alpha_3\alpha_5)\alpha_7^{-15} }}, |
we have the family of representatives
\langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle. |
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\nabla_3+\beta\nabla_4 \rangle ^{O(\alpha,\beta) = O(-\eta_3\alpha,\beta) = O(-\eta_3\alpha,-\beta) = O(\eta_3^2\alpha,-\beta) = O(\eta^2_3\alpha,\beta) = O(\alpha,-\beta)}, \\ \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+ \\ \nabla_7 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(-\eta_{13}^{11}\alpha,\eta^{10}_{13}\beta,\eta^6_{13}\gamma,\eta^4_{13}\mu) = \\ O(-\eta^9_{13}\alpha,-\eta^7_{13}\beta,\eta^{12}_{13}\gamma,\eta^8_{13}\mu) = O(-\eta^7_{13}\alpha,\eta^4_{13}\beta,-\eta^5_{13}\gamma,\eta^{12}_{13}\mu) = \\ O(-\eta^5_{13}\alpha,-\eta^{1}_{13}\beta,-\eta^{11}_{13}\gamma,-\eta^{3}_{13}\mu) = O(-\eta^{3}_{13}\alpha,-\eta^{11}_{13}\beta,\eta^4_{13}\gamma,-\eta^7_{13}\mu) = \\ O(-\eta^{1}_{13}\alpha,\eta^8_{13}\beta,\eta^{10}_{13}\gamma,-\eta^{11}_{13}\mu) = O(\eta^{12}_{13}\alpha,-\eta^5_{13}\beta,-\eta^{3}_{13}\gamma,\eta^{2}_{13}\mu) = \\ O(\eta^{10}_{13}\alpha,\eta^{2}_{13}\beta,-\eta^{9}_{13}\gamma,\eta^6_{13}\mu) = O(\eta^8_{13}\alpha,\eta^{12}_{13}\beta,\eta^{2}_{13}\gamma,\eta^{10}_{13}\mu) = \\ O(\eta^6_{13}\alpha,-\eta^{9}_{13}\beta,\eta^{8}_{13}\gamma,-\eta^{1}_{13}\mu) = O(\eta^4_{13}\alpha,\eta^{6}_{13}\beta,-\eta^{1}_{13}\gamma,-\eta^5_{13}\mu) = \\ O(\eta^{2}_{13}\alpha,-\eta^{3}_{13}\beta,-\eta^{7}_{13}\gamma,-\eta^9_{13}\mu) \end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(-\eta_3\alpha,\eta^2_3\beta,\eta^2_9\gamma) = O(\eta^2_3\alpha,-\eta_3\beta,\eta^4_9\gamma) = \\ O(\alpha,\beta,\eta^2_3\gamma) = O(-\eta_3\alpha,\eta^2_3\beta,\eta^8_9\gamma) = O(\eta^2_3\alpha,-\eta_3\beta,-\eta_9\gamma) = \\ O(\alpha,\beta,-\eta_3\gamma) = O(-\eta_3\alpha,\eta^2_3\beta,-\eta^5_9\gamma) = O(\eta^2_3\alpha,-\eta_3\beta,-\eta^7_9\gamma) \end{array}}}, \\ \langle \nabla_2+\nabla_3+\alpha\nabla_4 \rangle^{O(\alpha) = O(i\alpha) = O(-\alpha) = (-i\alpha)}, \\ \langle \nabla_2+\alpha\nabla_3+ \beta\nabla_4+\nabla_5 \rangle^{ O(\alpha,\beta) = O(-\eta_5\alpha,\eta^4_5\beta) = O(\eta^2_5\alpha,-\eta^3_5\beta) = O(-\eta^3_5\alpha,\eta^2_5\beta) = O(\eta_5^4\alpha,-\eta_5\beta)},\\ \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_{11}^{10}\alpha,\eta^6_{11}\beta,\eta^4_{11}\gamma = \\ O(-\eta^9_{11}\alpha,-\eta_{11}\beta,\eta^38_{11}\gamma) = O(\eta^8_{11}\alpha,-\eta^7_{11}\beta,-\eta_{11}\gamma) = \\ O(-\eta^7_{11}\alpha,\eta^2_{11}\beta,-\eta^5_{11}\gamma) = O(\eta^{6}_{11}\alpha,\eta^8_{11}\beta,-\eta^9_{11}\gamma) = \\ O(-\eta^{5}_{11}\alpha,-\eta^3_{11}\beta,\eta^2_{11}\gamma) = O(\eta^4_{11}\alpha,-\eta^9_{13}\beta,\eta^{6}_{11}\gamma) = \\ O(-\eta^3_{11}\alpha,\eta^{4}_{11}\beta,\eta^{10}_{11}\gamma) = O(\eta^2_{11}\alpha,\eta^{10}_{11}\beta,-\eta^{3}_{11}\gamma) = \\ O(-\eta_{11}\alpha,-\eta^{5}_{11}\beta,-\eta^{7}_{11}\gamma) \end{array}}}, \langle \nabla_3 \rangle, \\ \langle \nabla_3+\nabla_4 \rangle, \langle \nabla_3+ \alpha\nabla_4+\nabla_5 \rangle, \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle^{O(\alpha,\beta) = O(\alpha,-\eta_3\beta) = O(\alpha,\eta^2_3\beta)}, \\ \langle \nabla_4+\nabla_5 \rangle \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle^{{ \begin{array}{l}O(\alpha) = O(-\eta_3\alpha) = \\O(\eta^2_3\alpha) \end{array}}}, \\ \langle \nabla_4+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle^{ { \begin{array}{l} O(\alpha,\beta) = O(-\eta_5\alpha,\eta^4_5\beta) = O(\eta^2_5\alpha,-\eta^3_5\beta) = \\ O(-\eta^3_5\alpha,\eta^2_5\beta) = O(\eta^4_5\alpha,-\eta_5 \beta) \end{array}}}, \langle \nabla_5 \rangle, \langle \nabla_5+\nabla_6 \rangle, \\ \langle \nabla_5+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta^2_3\alpha)}, \langle \nabla_6 \rangle, \langle \nabla_6+\nabla_7 \rangle, \langle \nabla_7 \rangle,\end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{383}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{384}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_2e_2 = e_3 & e_2e_3 = \beta e_5 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{385}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{386}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{387}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{388}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{389} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_3e_3 = e_4 \\ {\mathbf{N}}_{390} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{391}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{392}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{393} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{394}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{395}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{396} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{397} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{398}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{399} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{400} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{401} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{10} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_4 \\ e_2e_2 = e_3 \\ e_3e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{10}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(1,2),(2,2)\} \end{array} & \begin{array}{l} \phi_k = \begin{pmatrix} \eta^k&0&0&0\\ 0& \eta^{2k}&0&0\\ 0&0& \eta^{4k}&0\\ t&0&0& \eta^{8k} \end{pmatrix} \\ {\eta = -\eta_5, \ k = 0,1,2,3,4} \end{array}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi_k^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&0&\alpha_3&\alpha_4\\ \alpha_1&\alpha_3&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi_k = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1&\alpha^*_2\\ \alpha^{**}&0&\alpha^*_3&\alpha^*_4\\ \alpha^{*}_1&\alpha^*_3&\alpha^*_5+\alpha^{**}&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix}, |
we have
\begin{array}{llll} \alpha_1^* = \eta^{4 k} (\eta^k \alpha_1+t \alpha_6), & \alpha_2^* = \eta^{8 k} (\eta^k \alpha_2+t \alpha_7), & \alpha_3^* = \eta^{6 k} \alpha_3, & \alpha_4^* = \eta^{10 k} \alpha_4, \\ \alpha_5^* = -t \eta^{2 k} \alpha_4+\eta^{8 k} \alpha_5, & \alpha_6^* = \eta^{12 k} \alpha_6, & \alpha_7^* = \eta^{16 k} \alpha_7. \end{array} |
Since
\langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_5\rangle; |
\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_3+\nabla_4 \rangle; |
\langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle; |
\langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle. |
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_3+\nabla_4 \rangle^{ {\begin{array}{l} O(\alpha, \beta, \gamma) = O(\alpha, \eta^4_5\beta, -\eta_5 \gamma) = O(\alpha, -\eta^3_5\beta, \eta^2_5 \gamma) = \\ O(\alpha, \eta^2_5\beta, -\eta^3_5 \gamma) = O(\alpha, -\eta_5\beta, \eta^4_5 \gamma)\end{array}}}, \\ \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_5 \rangle^{ {\begin{array}{l} O(\alpha, \beta, \gamma) = O(-\eta_5\alpha, \eta^2_5\beta, \eta^4_5 \gamma) = O(\eta^2_5\alpha, \eta^4_5\beta, -\eta^3_5 \gamma) = \\ O(-\eta^3_5\alpha, -\eta_5\beta, \eta^2_5 \gamma) = O(\eta^4_5\alpha, -\eta^3_5\beta, -\eta_5 \gamma) \end{array}}}, \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+ \\ \nabla_7 \rangle^{{\begin{array}{l} O(\alpha, \beta, \gamma, \mu, \nu) = O(\eta^4_5\alpha, \beta, \eta^4_5 \gamma, \eta^2_5\mu, -\eta_5\nu) = \\ O(-\eta^3_5\alpha, \beta, -\eta^3_5 \gamma, \eta^4_5\mu, \eta^2_5\nu) = O(\eta^2_5\alpha, \beta, \eta^2_5 \gamma,- \eta_5\mu, -\eta^3_5\nu) = \\ O(-\eta_5\alpha, \beta, -\eta_5 \gamma, -\eta^3_5\mu, \eta^4_5\nu) \end{array}}}, \\ \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+ \\ \nabla_6 \rangle^{{\begin{array}{l} O(\alpha, \beta, \gamma, \mu) = O(\eta_5^2\alpha, \eta_5^4\beta, -\eta_5^3 \gamma, -\eta_5\mu) = O(\eta_5^4\alpha, -\eta_5^3\beta, -\eta_5\gamma, \eta^2_5\mu) = \\ O(-\eta_5\alpha, \eta_5^2\beta, \eta_5^4 \gamma, -\eta^3_5\mu) = O(-\eta_5^3\alpha, -\eta_5\beta, \eta_5^2 \gamma, \eta^4_5\mu) \end{array}}},\end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{402}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{403}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = e_4+\gamma e_5 \\ {\mathbf{N}}_{404}^{\alpha, \beta,\gamma,\mu,\nu} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 \\ & & e_3e_3 = e_4+\mu e_5 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{405}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4+\mu e_5 & e_3e_4 = e_5 \\ \end{array} |
Here we will collect all information about
\begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{11}(\lambda) & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = \lambda e_4 \\ e_2e_2 = e_3\\ e_2e_3 = e_4 \\ e_3e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{11}(\lambda)) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(2,2),(3,3) \} \end{array} & \phi = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ t&0&0&1 \end{pmatrix}\\ \hline \end{array} |
Let us use the following notations:
\begin{array}{lll l} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array} |
Take
\phi^T\begin{pmatrix} 0&\alpha_1&\alpha_2&\alpha_3\\ \alpha_1&0&\alpha_4&\alpha_5\\ \alpha_2&\alpha_4&0&\alpha_6\\ \alpha_3&\alpha_5&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha_1^{*}&\alpha^{*}_2&\alpha^*_3\\ \alpha_1^{*}&0&\alpha^*_4&\alpha^*_5\\ \alpha^{*}_2&\alpha^*_4&0&\alpha^*_6\\ \alpha^*_3&\alpha^*_5&\alpha^*_6&\alpha^*_7 \end{pmatrix} |
we have
\begin{array}{llll} \alpha_1^* = \alpha_1+\alpha_5t, & \alpha_2^* = \alpha_2+\alpha_6t, & \alpha_3^* = \alpha_3+\alpha_7t, & \alpha_4^* = \alpha_4, \\ \alpha_5^* = \alpha_5, & \alpha_6^* = \alpha_6, & \alpha_7^* = \alpha_7. \end{array} |
Since
\langle \alpha\nabla_1+\beta\nabla_2+\nabla_3+\gamma\nabla_4 \rangle; |
\langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\nabla_5 \rangle; |
\langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle; |
\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle. |
Summarizing, we have the following distinct orbits:
\begin{array}{c} \langle \alpha\nabla_1+\beta\nabla_2+\nabla_3+\gamma\nabla_4 \rangle,\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle, \\\langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle,\langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\nabla_5 \rangle,\end{array} |
which gives the following new algebras:
\begin{array}{llllllllllllllllll} {\mathbf{N}}_{406}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4+\alpha e_5 & e_1e_3 = \beta e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_4+\gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{407}^{\lambda,\alpha, \beta,\gamma,\mu,\nu} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4+\alpha e_5 & e_1e_3 = \beta e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_4+\gamma e_5 & e_2e_4 = \mu e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{408}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4+\alpha e_5 & e_1e_4 = \beta e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4+\gamma e_5 & e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{409}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_4+\gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \end{array} |
Remark 2. Note that the algebras
Theorem 5.1. Let
\begin{array}{llllllll} {\mathbf N}_{01} & : & e_1 e_1 = e_2 & e_1 e_2 = e_3 & e_2e_3 = e_4 \\ {\mathbf N}_{02} & : & e_1 e_1 = e_2 & e_1 e_2 = e_3 & e_1e_3 = e_4 & e_2 e_3 = e_4 && \\ {\mathbf N}_{03} & : & e_1 e_1 = e_2 & e_1 e_2 = e_3 & e_3 e_3 = e_4 &&\\ {\mathbf N}_{04} & : & e_1 e_1 = e_2 & e_1 e_2 = e_3 & e_2e_2 = e_4 & e_3 e_3 = e_4 &&\\ {\mathbf N}_{05} & : & e_1 e_1 = e_2 & e_1 e_3 = e_4 & e_2 e_2 = e_3 && \\ {\mathbf N}_{06} & : & e_1 e_1 = e_2 & e_1e_2 = e_4 & e_1 e_3 = e_4 & e_2 e_2 = e_3 && \\ {\mathbf N}_{07} & : & e_1 e_1 = e_2 & e_2 e_2 = e_3 & e_2 e_3 = e_4 && \\ {\mathbf N}_{08} & : & e_1 e_1 = e_2 & e_1e_3 = e_4 & e_2 e_2 = e_3 & e_2 e_3 = e_4 && \\ {\mathbf N}_{09} & : & e_1 e_1 = e_2 & e_2 e_2 = e_3 & e_3 e_3 = e_4 && \\ {\mathbf N}_{10} & : & e_1 e_1 = e_2 & e_2e_2 = e_3 & e_1e_2 = e_4 & e_3 e_3 = e_4 &&\\ {\mathbf N}_{11}^{\lambda} & : & e_1 e_1 = e_2 & e_1e_2 = \lambda e_4 & e_2 e_2 = e_3 \\ && e_2e_3 = e_4 & e_3 e_3 = e_4 &\\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{12} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{13}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_4 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{14} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{15} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{16} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{17} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{18} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{19} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_3 & e_3e_3 = e_5 \\ {\mathbf{N}}_{20}^{\alpha} & : & e_1e_1 = e_2 & { e_1e_2 = e_4+\alpha e_5 }& e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{21} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{22}^{\alpha\neq 1} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{23}^{\alpha, \beta} & : & e_1e_1 = e_2 & { e_1e_2 = \beta e_4 +\alpha e_5 }& e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{24}^{\alpha} & : & e_1e_1 = e_2 & { e_1e_2 = \alpha e_4+e_5 }& e_2e_2 = e_3 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{25} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{26} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 \\ && e_2e_2 = e_3 & e_3e_3 = e_4 \\ {\mathbf{N}}_{27} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{28} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{29} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_3 = e_5 \\ {\mathbf{N}}_{30} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{31} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{32}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{33} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{34} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{35} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{36} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{37} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{38}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = \alpha e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{39} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{40} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{41} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{42} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{43} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{44} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{45} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{46} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_4 = e_5 \\ {\mathbf{N}}_{47} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_5 \\ && e_2e_2 = e_3 & e_4e_4 = e_5 \\ {\mathbf{N}}_{48} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_3e_3 = e_5 \\ {\mathbf{N}}_{49}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{50}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_2e_2 = e_3 & e_2e_3 = \beta e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{51}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{52} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{53}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{54} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_3 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{55} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_3 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{56} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_3 & e_3e_4 = e_5 \\ {\mathbf{N}}_{57} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{58} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{59} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ {\mathbf{N}}_{60} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 \\ & & e_2e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{61} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_4e_4 = e_5 \\ {\mathbf{N}}_{62} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{63}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{64} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{65} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_3e_3 = e_5 \\ {\mathbf{N}}_{66} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{67} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{68} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{69} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{70} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{71} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{72} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ && e_1e_4 = \frac{3}{4}e_5 & e_2e_2 = e_4 & e_2e_3 = -\frac{3}{4}e_5 \\ {\mathbf{N}}_{73}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = 3e_5 & e_2e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{74}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ && e_1e_4 = \alpha e_5 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{75}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ && e_2e_2 = e_4 & e_2e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{76}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & { e_1e_4 = (1+\alpha) e_5 } \\ && e_2e_2 = e_4 & e_2e_3 = 3\alpha e_5 \\ {\mathbf{N}}_{77}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_3 = 3e_5 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{78}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = 3\alpha e_5 & e_2e_4 = \beta e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{79} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{80}^{\alpha \neq 1} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_4 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{81}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{82} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{83}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{84} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 & e_2e_3 = e_4 \\ {\mathbf{N}}_{85}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{86}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{87} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{88}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{89}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{90}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{91} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{92} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{93} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{94} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{95}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{96}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{97} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{98} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{99}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{100} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{101} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{102} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_3 = e_4 \\ {\mathbf{N}}_{103} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ {\mathbf{N}}_{104}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{105}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{106} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{107} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{108} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{109} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{110} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{111} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{112}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{113} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{114} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{115} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{116} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{117} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{118} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{119} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{120} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{121} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{122} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{123} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{124} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}^{\alpha}_{125} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}^{\alpha,\beta}_{126} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_2e_2 = e_5 \\ & & e_3e_3 = e_4 + \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}^{\alpha}_{127} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 \\ && e_2e_3 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{128} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 \\ && e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{129} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{130} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{131} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{132} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_3e_3 = e_4+e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{133} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{134} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{135}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_3 = -2e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{136}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 \\ && e_2e_2 = e_4+\alpha e_5 & e_2e_3 = \beta e_5 \\ && e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{137}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = \alpha e_5 & e_2e_4 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{138}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4+e_5 & e_2e_4 = e_5 & e_3e_3 = 4e_5 \\ {\mathbf{N}}_{139}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{140}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{141}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{142}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4+e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{143}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_3 = e_5 & e_2e_4 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{144}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{145}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{146} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{147} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{148} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{149} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{150}^{\alpha, \beta, \gamma, \mu } & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4+\mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{151}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4+\beta e_5 \\ {\mathbf{N}}_{152}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{153}^{\alpha, \beta, \gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & { e_3e_3 = e_4+\beta e_5 } & e_3e_4 = \gamma e_5 \\ {\mathbf{N}}_{154}^{\alpha, \beta, \gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & { e_3e_3 = e_4+\gamma e_5 } & e_4e_4 = e_5 \\ {\mathbf{N}}_{155} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_3e_3 = e_4 \\ {\mathbf{N}}_{156}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_2e_3 = e_5 \\ & & { e_3e_3 = e_4+\alpha e_5 } \\ {\mathbf{N}}_{157}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{158}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4+\alpha e_5 & e_3e_4 = \beta e_5 \\ {\mathbf{N}}_{159} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ & & e_3e_3 = e_4+e_5 \\ {\mathbf{N}}_{160}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4+\beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{161} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_3 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{162}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ && e_3e_3 = e_4+e_5 & e_3e_4 = \alpha e_5 \\ {\mathbf{N}}_{163}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4+\alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{164}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 \\ {\mathbf{N}}_{165} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4+e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{166} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{167} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{168}^{\lambda \neq 1; 2} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_1e_4 = (\lambda-4)e_5 & { e_2e_2 = \lambda e_4 + 4(1-\lambda)(\lambda-2)e_5 } \\ && { e_2e_3 = - \lambda(\lambda+2)e_5 }\\ {\mathbf{N}}_{169}^{\alpha\neq0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = -2 e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{170}^{\lambda, \alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_1e_4 = \alpha e_5 & e_2e_2 = \lambda e_4 & { e_2e_3 = (1+\alpha(3\lambda-2)) e_5 } \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{171}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_1e_4 = e_5 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = (3\lambda-2) e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{172}^{\lambda \neq 0,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 + e_5 & e_2e_3 = \alpha e_5 & e_2e_4 = \frac{\lambda} {4} e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{173}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{174}^{\lambda\neq-2, \alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 + \alpha e_5 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{175}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_2e_2 = -2 e_4+\alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{176}^{\lambda,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4+\alpha e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{177}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4+ e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{178}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_2e_2 = \lambda e_4 & e_2e_3 = e_5 & e_2e_4 = e_5 \\ {\mathbf{N}}_{179}^{\lambda,\alpha\neq0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{180}^{\lambda,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{181}^{\alpha \neq 0 } & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = -2 e_4 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{182} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = -2 e_4 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{183}^{\lambda\neq 2} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{184}^{\lambda, \alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{185}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{186}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{187}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{188}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{189}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{190} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\\end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{191}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{191}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{192} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{193} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{194} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{195} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{196} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ {\mathbf{N}}_{197} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ {\mathbf{N}}_{198} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{199}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{200} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{201} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{202}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{203} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{204} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{205} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{206}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{207}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{208} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{209}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_5 \\ && e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{210}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{211} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{212} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{213} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{214} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{215}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{216}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = e_4+\alpha e_5 & e_2e_3 = \beta e_5 \\ & & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{217}^{\alpha} & : & e_1e_1 = \alpha e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{218}^\alpha & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{219} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = 2e_5 \\ && e_2e_2 = e_4+e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{220}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4+\alpha e_5 & e_3e_3 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{221}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{222}^{\alpha\neq 0} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{223}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{224}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{225}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4+e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{226}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{227}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{228} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{229} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{230} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{231} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{232}^{\alpha, \beta, \gamma, \mu} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4+\beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{233}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{234}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4+e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{235}^{\alpha, \beta, \gamma} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{236}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{237}^{\alpha\neq0, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4+e_5 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{238}^{\alpha\neq0, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{239}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{240}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4+e_5 \\ & & e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{241} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{242}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ & & e_3e_3 = e_5 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{243} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{244} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{245} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{246}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{247}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{248} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{249}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{250} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_3 = e_5 & e_3e_3 = e_4 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{251}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{252}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{253}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{254} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{255}^{\alpha} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{256}^{\alpha} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{257} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{258} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{259} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{260}^{\alpha} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{261} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_4 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{262} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{263} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{264} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{265} & : & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{266} & : & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{267} & : & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{268} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{269} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{270} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{271}^{\alpha, \beta, \gamma,\mu} & : & e_1e_1 = e_4+e_5 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_2e_2 = \beta e_5 \\ && e_2e_3 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{272}^{\alpha, \beta} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{273}^{\alpha, \beta} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{274}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_4+e_5 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = \beta e_5 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{275}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{276}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{277}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{278} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{279}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{280}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{281} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{282}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{283}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{284}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{285}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{286} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_3e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{287} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{288}^{\alpha, \beta, \gamma,\mu,\nu} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 \\ && e_2e_2 = e_4+\gamma e_5 & e_2e_3 = \mu e_5 & e_3e_3 = e_4 \\ & & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{289}^{\alpha, \beta, \gamma,\mu} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_4 = \beta e_5 \\ && e_2e_2 = e_4+\gamma e_5 & e_2e_4 = \mu e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{290}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{291}^{\alpha, \beta\neq0 ,\gamma} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ & & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{292}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ & & e_1e_4 = \alpha e_5 & e_2e_2 = \beta e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = -\alpha e_5 \\ {\mathbf{N}}_{293}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = e_4 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{294}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{295}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{296}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ & & e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{297}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{298}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ & & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{299}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{300}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_4 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{301} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{302} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{303}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{304}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{305} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{306} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{307} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{308} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{309}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4+\beta e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{310}^{\alpha, \beta\neq-1} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = e_4 & e_3e_3 = \beta e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{311}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{312}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4+\beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{313}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{314}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{315}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = \beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{316}^{\alpha, \beta, \gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{317} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{318}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{319}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{320}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{321} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ & & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{322}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{323}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{324}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{325} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{326}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{327}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{328} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{329} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}^\alpha_{330} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{331} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{332} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{333} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{334}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{335}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ && e_2e_2 = e_4+\gamma e_5 & e_2e_3 = \mu e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{336}^{\alpha, \beta,\gamma, \mu, \nu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 \\ & & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = \mu e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{337}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = \mu e_5 \\ && e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{338}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = -2e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{339}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{340}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{341}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{342}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{343}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{344}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{345}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{346}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{347}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{348} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ {\mathbf{N}}_{349} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{350} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{351}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{352} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{353} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_4 = e_5 \\ {\mathbf{N}}_{354} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_4e_4 = e_5 \\ {\mathbf{N}}_{355}^{\alpha\neq0, \beta} & : & e_1e_1 = e_2 & { e_1e_3 = e_4+\alpha e_5 } \\ && e_1e_4 = e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & { e_3e_3 = (\beta+2)e_5 } \\ {\mathbf{N}}_{356}^{\alpha\neq0, \beta,\gamma} & : & e_1e_1 = e_2 & { e_1e_3 = e_4+\alpha e_5 } \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_2e_4 = e_5 \\ & & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{357}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & { e_1e_3 = e_4+\alpha e_5 } \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = \gamma e_5 \\ & & e_3e_4 = \mu e_5 & e_4e_4 = e_5 & \\ {\mathbf{N}}_{358}^{\alpha,\beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{359}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 & \\ {\mathbf{N}}_{360}^{\alpha,\beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{361}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = \beta e_5 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{362}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = \beta e_5 \\ & & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{363}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{364}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 \ & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{365}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{366}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_1e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{367} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ {\mathbf{N}}_{368}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{369}^{\alpha,\beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{370} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{371}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{372}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{373}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{374} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{375}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{376} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{377} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{378} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{379}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 \\ && e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_4+\beta e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{380}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 \\ & & e_1e_4 = \beta e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{381}^{\alpha, \beta,\gamma,\mu, \nu} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 \\ & & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{382}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{383}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{384}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{385}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{386}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{387}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{388}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{389} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_3e_3 = e_4 \\ {\mathbf{N}}_{390} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_3 = e_4 \\ \end{array} |
\begin{array}{llllllll} {\mathbf{N}}_{391}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{392}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{393} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{394}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ & & e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{395}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{396} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{397} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{398}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{399} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{400} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 \\ && e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{401} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{402}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{403}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = e_4+\gamma e_5 \\ {\mathbf{N}}_{404}^{\alpha, \beta,\gamma,\mu,\nu} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 \\ & & e_3e_3 = e_4+\mu e_5 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{405}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4+\mu e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{406}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & { e_1e_2 = \lambda e_4+\alpha e_5 } \\ && e_1e_3 = \beta e_5 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_4+\gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{407}^{\lambda,\alpha, \beta,\gamma,\mu,\nu} & : & e_1e_1 = e_2 & { e_1e_2 = \lambda e_4+\alpha e_5 } \\ && e_1e_3 = \beta e_5 & e_2e_2 = e_3 & { e_2e_3 = e_4+\gamma e_5 } \\ & & e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{408}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & { e_1e_2 = \lambda e_4+\alpha e_5 } \\ & & e_1e_4 = \beta e_5 & e_2e_2 = e_3 & e_2e_3 = e_4+\gamma e_5 \\ && e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{409}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4 & e_1e_3 = \alpha e_5 \\ && e_1e_4 = \beta e_5 & e_2e_2 = e_3 & { e_2e_3 = e_4+\gamma e_5 } \\ & & e_2e_4 = e_5 & e_3e_3 = e_4 \end{array} |
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