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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
$\phi \theta (x,y) = \theta \left( \phi \left( x\right) ,\phi \left( y\right) \right) $. |
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
$\dim (\operatorname{Ann}({\bf A})) = m\neq0$. |
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)]_{{\bf A}'}. $ |
Since
Definition 1.2. Let
Our task is to find all central extensions of an algebra
Let
$ W_{1} = \langle \left[ \theta _{1}\right] ,\left[ \theta _{2}\right] ,\dots, \left[ \theta _{s}\right] \rangle ,W_{2} = \langle \left[ \vartheta _{1}\right] ,\left[ \vartheta _{2}\right] ,\dots,\left[ \vartheta _{s}\right] \rangle \in G_{s}\left( {\rm H^{2}}\left( {\bf A},\mathbb C\right) \right), $ |
we easily have that if
$ {\bf T}_{s}({\bf A}) = \left\{ W = \langle \left[ \theta _{1}\right] , \dots,\left[ \theta _{s}\right] \rangle \in G_{s}\left( {\rm H^{2}}\left( {\bf A},\mathbb C\right) \right) : \bigcap\limits_{i = 1}^{s}\operatorname{Ann}(\theta _{i})\cap\operatorname{Ann}({\bf A}) = 0\right\}, $ |
which is stable under the action of
Now, let
$ {\bf E}\left( {\bf A},{\mathbb V}\right) = \left\{ {\bf A}_{\theta }:\theta \left( x,y\right) = \sum\limits_{i = 1}^{s}\theta _{i}\left( x,y\right) e_{i} \ \ {\rm{and}} \ \ \langle \left[ \theta _{1}\right] ,\left[ \theta _{2}\right] ,\dots, \left[ \theta _{s}\right] \rangle \in {\bf T}_{s}({\bf A}) \right\} . $ |
We also have the following result, which can be proved as in [10,Lemma 17].
Lemma 1.3. Let
$ \operatorname{Orb}\langle \left[ \theta _{1}\right] , \left[ \theta _{2}\right] ,\dots,\left[ \theta _{s}\right] \rangle = \operatorname{Orb}\langle \left[ \vartheta _{1}\right] ,\left[ \vartheta _{2}\right] ,\dots,\left[ \vartheta _{s}\right] \rangle. $ |
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 (
$ \theta(x,y) = \theta(y,x), $ |
$ \theta((xy)a,b)+\theta((xb)a,y)+\theta(x,(yb)a) = \theta((xy)b,a)+\theta((xa)b,y)+\theta(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
$ {\bf R}_{s}({\mathfrak{D}}) = \left\{ {\bf W}\in {\bf T}_{s}({\mathfrak{D}}) :{\bf W}\in G_{s}({\rm H_\mathfrak{D}^2}({\mathfrak{D}},{\mathbb C}) ) \right\}, $ |
$ {\bf U}_{s}({\mathfrak{D}}) = \left\{ {\bf W}\in {\bf T}_{s}({\mathfrak{D}}) :{\bf W}\notin G_{s}({\rm H_\mathfrak{D}^2}({\mathfrak{D}},{\mathbb C}) ) \right\}. $ |
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
$ \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{3*}_{02} & \begin{array}{l}e_1e_1 = e_2 \\ e_1e_2 = e_3 \end{array} & H2D(N3∗02)=⟨[Δ13],[Δ22]⟩,H2C(N3∗02)=H2D(N3∗02)⊕⟨[Δ23],[Δ33]⟩ & \phi = (x00yx20z2xyx3)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ13],∇2=[Δ22],∇3=[Δ23],∇4=[Δ33]. $ |
Take
$ \phi^T(00α10α2α3α1α3α4)\phi = (α∗α∗∗α∗1α∗∗α∗2α∗3α∗1α∗3α∗4), $ |
we have
$ \alpha_1^* = (\alpha_1x+\alpha_3y+\alpha_4z)x^3, \quad \alpha_2^* = (\alpha_2x^2+4\alpha_3xy+4\alpha_4y^2)x^2 ,\\ \alpha_3^* = (\alpha_3x+2\alpha_4y)x^4 , \qquad \alpha_4^* = \alpha_4x^6. $ |
We are interested only in
$ \theta_1 = \alpha_1\nabla_1+\alpha_2\nabla_2+\alpha_3\nabla_3+\alpha_4\nabla_4 \ \ \rm{and} \ \ \theta_2 = \beta_1\nabla_1+\beta_2\nabla_2+\beta_3\nabla_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 = 2 \alpha_4^2 \beta_2, $ $y = -\alpha_3 \alpha_4 \beta_2,$ $z = \alpha_3^2 (-2 \beta_1+\beta_2)+2 \alpha_4 (\alpha_2 \beta_1-\alpha_1 \beta_2),$ |
we have the representatives
$ x = 4\beta_3\alpha_4, y = -\beta_2\alpha_4, z = \beta_2\alpha_3-4\alpha_1\beta_3,$ |
we have the representative
$ x = \frac{4\beta_1-\beta_2}{4\beta_3}, y = \frac{\beta_2^2-4\beta_1\beta_2}{16\beta_3^2}, z = \frac{(4\beta_1-\beta_2)(8\beta_1\alpha_3\beta_3-4\beta_1\beta_2\alpha_4-8\alpha_1\beta_3^3+\beta_2^2\alpha_4)}{32\beta_3^3\alpha_4},$ |
we have the representative
$ x = \sqrt{\frac{4\alpha_2\beta_3^2-4\beta_2\alpha_3\beta_3+\beta_2^2\alpha_4}{4\beta_3^2\alpha_4}},$ $ y = -\frac{ \beta_2\sqrt{\alpha_4 \beta_2^2-4 \alpha_3 \beta_2 \beta_3+4 \alpha_2 \beta_3^2}}{8\beta_3^2\sqrt{\alpha_4}}, $ $z = \frac{(8\beta_1\alpha_3\beta_3-4\beta_1\beta_2\alpha_4-8\alpha_1\beta_3^3+\beta_2^2\alpha_4)\sqrt{4\alpha_2\beta_3^2-4\beta_2\alpha_3\beta_3+\beta_2^2\alpha_4}}{16\beta_3^3\alpha_4\sqrt{\alpha_4}},$ |
we have the family of representatives
Summarizing, we have the following distinct orbits:
$ \langle \nabla_1, \nabla_2+\nabla_4 \rangle, \, \langle \nabla_1+4\nabla_2, -24(\nabla_2+\nabla_3) \rangle, \, \langle \nabla_1+\lambda\nabla_2, \nabla_3 \rangle, \, \langle \nabla_1 +\lambda \nabla_2, \nabla_4\rangle, \\ \langle \alpha\nabla_1+\nabla_3, \nabla_2+\nabla_4 \rangle, \, \langle \nabla_1+\nabla_3, \nabla_4 \rangle, \, \langle \nabla_2, -3 \nabla_3 \rangle, \, \langle \nabla_2, \nabla_4 \rangle, \, \langle \nabla_3, \nabla_4 \rangle. $ |
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
$ \begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{3*}_{04} & \begin{array}{l}e_1e_1 = e_2 \\ e_2e_2 = e_3 \end{array} & H2D(N3∗04)=⟨[Δ12]⟩,H2C(N3∗04)=H2D(N3∗04)⊕⟨[Δ13],[Δ23],[Δ33]⟩& \phi = (x000x20z0x4)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ12],∇2=[Δ13],∇3=[Δ23],∇4=[Δ33]. $ |
Take
$ \phi^T(0α1α2α10α3α2α3α4)\phi = (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:
$ \theta_1 = \alpha_1\nabla_1+\alpha_2\nabla_2+\alpha_3\nabla_3+\alpha_4\nabla_4 \ \ \rm{and} \ \ \theta_2 = \beta_1\nabla_1+\beta_2\nabla_2+\beta_3\nabla_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 = \sqrt{{\alpha_3}{\alpha_4}^{-1}}\ {\rm{and}}\ z = -{\alpha_1}\sqrt{{\alpha_3}^{-1} \alpha_4^{-1}},$ |
we have the representative
$ x = \sqrt{\frac{\beta_1}{\beta_2}} {\text{ and }} z = \frac{(\alpha_1\beta_2-\beta_1\alpha_2)\sqrt{\beta_1}}{(\beta_1\alpha_4-\beta_2\alpha_3)\sqrt{\beta_2}},$ |
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:
$ \langle \nabla_1, \nabla_2 \rangle, $ $ \langle \nabla_1, \nabla_2+\nabla_3 \rangle,$ $ \langle \nabla_1, \nabla_3 \rangle,$ $ \langle \nabla_1, \nabla_3+\nabla_4 \rangle, $ $ \langle \nabla_1, \nabla_4 \rangle,$ $\langle \nabla_1+\nabla_2, \alpha\nabla_1+\nabla_3+\nabla_4 \rangle^{O(\alpha) = O(-\alpha)}, $ $ \langle \nabla_1+\nabla_2,\nabla_3 \rangle, $ $ \langle \nabla_1+\nabla_2, \alpha \nabla_3 + \nabla_4 \rangle_{\alpha\neq 1},$ $ \langle \beta\nabla_1+\nabla_2+\nabla_3, \alpha \nabla_1 + \nabla_4 \rangle, $ $ \langle \alpha\nabla_1+\nabla_3, \nabla_1+\nabla_4 \rangle^{O(\alpha) = O(-\eta_3 \alpha) = O(\eta_3^2 \alpha)}, $ $ \langle \nabla_1+\nabla_3, \nabla_4 \rangle,$ $\langle \nabla_1+\nabla_4, \nabla_2 \rangle,$ $ \langle \nabla_2, \nabla_3 \rangle, $ $ \langle \nabla_2, \nabla_3+\nabla_4 \rangle, $ $ \langle \nabla_2, \nabla_4 \rangle,$ $ \langle \nabla_3, \nabla_4 \rangle.$ |
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
$ \begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{02} & \begin{array}{l} e_1e_1 = e_2\\ e_1e_2 = e_3 \end{array} & H2D(N4∗02)=⟨[Δ13],[Δ22],[Δ14],[Δ24],[Δ44]⟩H2C(N4∗02)=H2D(N4∗02)⊕⟨[Δ23],[Δ33],[Δ34]⟩ & \phi = (x000qx200w2xqx3re00t)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ13],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. $ |
Take
$ \phi^T(00α1α20α3α4α5α1α4α6α7α2α5α7α8)\phi = (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 = \sqrt[4]{\alpha_2\alpha_4-\alpha_1\alpha_5}, t = \alpha_4^2, r = -\alpha_4\alpha_5, q = -\frac{\alpha_1\sqrt[4]{\alpha_2\alpha_4-\alpha_1\alpha_5}}{\alpha_4}, $ |
we have the representative
$ x = \frac{\alpha_3-4\alpha_1}{\alpha_4}, t = \frac{(\alpha_3-4\alpha_1)^4}{\alpha_4^2(\alpha_2\alpha_4-\alpha_1\alpha_5)}, r = \frac{\alpha_5(\alpha_3-4\alpha_1)^4}{\alpha_4^3(\alpha_1\alpha_5-\alpha_2\alpha_4)}, q = \frac{4\alpha^2_1-\alpha_1\alpha_3}{\alpha^2_4},$ |
we have the representative
$ x = \alpha_4\alpha_8, t = \alpha^3_4\alpha^2_8, q = -\alpha_1\alpha_8, r = -\alpha_4^2\alpha_5\alpha_8^2, e = \alpha_1\alpha_5-\alpha_2\alpha_4, $ |
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,$ $q = -\frac{\alpha_5}{2\alpha_7},$ $ e = - \frac{\alpha_1}{\alpha_7},$ $w = \frac{\alpha_5^2+2\alpha_1\alpha_8-2\alpha_2\alpha_7}{2 \alpha_7^2},$ $t = \frac{\alpha_3}{\alpha_7},$ $ r = -\frac{\alpha_3\alpha_8}{2\alpha_7^2}, $ |
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.
$x = 1, t = \frac{\alpha_6^2}{\alpha_2\alpha_6-\alpha_1\alpha_7}, q = -\frac{\alpha_4}{2\alpha_6}, r = \frac{\alpha_6\alpha_7}{\alpha_1\alpha_7-\alpha_2\alpha_6}, e = 0, w = \frac{\alpha^2_4-2\alpha_1\alpha_6}{\alpha_6},$ |
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 = \frac{\alpha_6^2}{\alpha_5\alpha_6-\alpha_4\alpha_7}x^4,$ $q = -\frac{\alpha_4}{2\alpha_6}x,$ $r = \frac{\alpha_6\alpha_7}{\alpha_4\alpha_7-\alpha_5\alpha_6}x^4,$ $e = 0,$ $w = \frac{\alpha^2_4-2\alpha_1\alpha_6}{\alpha_6}x,$ |
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
$ \begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{04} & \begin{array}{l} e_1e_1 = e_2 \\ e_2e_2 = e_3 \end{array} & H2D(N4∗04)=⟨[Δ12],[Δ14],[Δ24],[Δ44]⟩,H2C(N4∗04)=H2D(N4∗04)⊕⟨[Δ13],[Δ23],[Δ33],[Δ34]⟩ & \phi = (x0000x200y0x4rz00t)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ12],∇2=[Δ13],∇3=[Δ14],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. $ |
Take
$ \phi^T(0α1α2α3α10α4α5α2α4α6α7α3α5α7α8)\phi = (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 = \alpha_2\alpha_5,$ $t = \alpha_2^4\alpha_5^2,$ $z = -\alpha_1\alpha_2$, $r = -\alpha_2^3\alpha_3\alpha_5^2,$ $y = 0,$ |
we have the representative
$x = \alpha_3,$ $t = {\sqrt{\alpha_2}}\alpha_8^2,$ $z = -{\alpha_3},$ $r = 0,$ $y = 0,$ |
we have the representative
$ x = \sqrt{{\alpha_1}{\alpha_2}^{-1}},$ $t = \sqrt[4]{\alpha_1^5\alpha_2^{-3}}\sqrt{\alpha_8^{-1}},$ $z = -\sqrt{\alpha_1\alpha_2^{-1}} \alpha_3\alpha_8^{-1},$ $r = 0,$ $y = 0,$ |
we have the representative
$ x = {\frac{\alpha^2_5}{\alpha_2\alpha_8}}, t = {\frac{\alpha_5^5}{\alpha_2^2\alpha^3_8}}, z = -{\frac{\alpha_1\alpha_5}{\alpha_2\alpha_8}}, r = \frac{\alpha_5^4(\alpha_1\alpha_8-\alpha_3\alpha_5)}{\alpha_2^3\alpha_8^3}, 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
$x = \sqrt{\alpha_4 \alpha_6^{-1}},$ $t = \alpha_4^2 \sqrt{ \alpha_6^{-3} \alpha_8^{-1}},$ $z = -\alpha_3{\alpha_8^{-1}}\sqrt{\alpha_4 \alpha_6^{-1}},$ $y = 0,$ $r = 0,$ |
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
$ \begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{08} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_3 \\ e_2e_2 = e_4 \end{array} & H2D(N4∗08)=⟨[Δ13],[Δ14]+3[Δ23]⟩H2C(N4∗08)=H2D(N4∗08)⊕⟨[Δ14],[Δ24],[Δ33],[Δ34],[Δ44]⟩ & \phi = (x000yx200z2xyx30ty2x2yx4)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ13],∇2=[Δ14]+3[Δ23],∇3=[Δ14],∇4=[Δ24],∇5=[Δ33],∇6=[Δ34],∇7=[Δ44]. $ |
Take
$ \phi^T(00α1α2+α3003α2α4α13α2α5α6α2+α3α4α6α7)\phi = (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
$ \langle -\frac{1}{4}\nabla_2+\nabla_3 \rangle {\text{ and }} \langle \nabla_1-\frac{1}{4}\nabla_2+\nabla_3 \rangle $ |
depending on
2.
3.
$ x = \frac{\alpha_2+\alpha_3}{\alpha_5},$ $y = \frac{3\alpha_2\alpha_3+3\alpha_3^2}{2\alpha_5^2},$ $z = -\frac{(\alpha_2+\alpha_3)(2\alpha_1\alpha_5+12\alpha_2\alpha_3+3\alpha_3^2)}{4\alpha_5^3},$ |
we have the representative
$ x = 2(\alpha_4-\alpha_5), y = 3\alpha_3, z = 0, t = 0, $ |
we have the representative
$ x = \frac{2(\alpha_2\alpha_5-\alpha_2\alpha_4+\alpha_3\alpha_5)+\alpha_3\alpha_4}{2(\alpha_5^2-\alpha_4\alpha_5)},y = \frac{3\alpha_3(2(\alpha_2\alpha_5-\alpha_2\alpha_4+\alpha_3\alpha_5)+\alpha_3\alpha_4)}{2\alpha_5(\alpha_5-\alpha_4)^2}, z = \\ -\frac{(2 \alpha_2 (\alpha_4-\alpha_5)-\alpha_3 (\alpha_4+2 \alpha_5)) (4 \alpha_1 (\alpha_4-\alpha_5)^2-24 \alpha_2 \alpha_3 (\alpha_4-\alpha_5)+3 \alpha_3^2 (\alpha_4+2 \alpha_5))}{8 (\alpha_4-\alpha_5)^3 \alpha_5^2}, t = 0, $ |
we have the family of representatives
4. if
$z = y^2 - \frac{ \alpha_3}{\alpha_6} + \frac{2 y ( \alpha_5-\alpha_4)}{3 \alpha_6}{\text{ and }}t = -\frac{x^2 \alpha_1 + x y (4 \alpha_2 + \alpha_3) + x z \alpha_5 + y (y \alpha_4 + z \alpha_6)}{ \alpha_6)}, $ |
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
$ \begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{09} & \begin{array}{l} e_1e_1 = e_2 \\ e_2e_3 = e_4 \end{array} & H2D(N4∗09)=⟨[Δ12],[Δ13],[Δ22],[Δ33]⟩H2C(N4∗09)=H2D(N4∗09)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩& \phi = (x0000x20000r0t0sx2r)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ12],∇2=[Δ13],∇3=[Δ14],∇4=[Δ22],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. $ |
Take
$ \phi^T(0α1α2α3α1α40α5α20α6α7α3α5α7α8)\phi = (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 = \alpha_1 \alpha_4^{-1},$ $r = \alpha_1\alpha_3^{-1},$ $s = -\alpha_1\alpha_2 \alpha^{-2}_3,$ $ t = 0,$ |
we have the family of representatives
2.
$x = {\sqrt[4]{\alpha_4\alpha_6\alpha_5^{-2}}},$ $r = {\alpha_4}{\alpha_5}^{-1},$ $t = -{\alpha_1\sqrt[4]{\alpha_4\alpha_6 \alpha_5^{-6}}}, s = 0,$ |
we have the representative
$ r = 1, x = \sqrt[3]{{\alpha_2}{\alpha_5}^{-1}}, t = -\alpha_1\sqrt[3]{\alpha_2\alpha_5^{-4}}, s = 0,$ |
we have the representative
$ x = \sqrt[3]{{\alpha_2}{\alpha_5}^{-1}},$ $r = \alpha_6^{-1}\sqrt[3]{\alpha_2^{4}\alpha_5^{-1}},$ $t = -\alpha_1\sqrt[3]{\alpha_2\alpha_5^{-4}}, s = 0,$ |
we have the representative
$ x = \sqrt[3]{\alpha_2\alpha_5^{-1}}, r = {\alpha_4}{\alpha_5}^{-1}, t = -\alpha_1\sqrt[3]{\alpha_2 \alpha_5^{-4}}, s = 0,$ |
we have the family of representatives
$ x = {\alpha_3}{\alpha_5}^{-1}, r = {\alpha_3^4}{\alpha_5^{-3}\alpha_6^{-1}}, s = -{\alpha_2\alpha_3^3}{\alpha_5^{-3}\alpha_6^{-1}}, t = -{\alpha_1\alpha_3}{\alpha_5^{-2}},$ |
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} & ϕ=(x000yx2−zrx0z0r0tz2+2xysx3),r2=x3\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ13],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. $ |
Take
$ \phi^T(00α1α20α3α4α5α1α4α6α7α2α5α7α8)\phi = (α∗α∗∗α∗1α∗2α∗∗α∗3α∗4α∗5α∗1α∗4α∗6+α∗∗α∗7α∗2α∗5α∗7α∗8), $ |
we have
$ α∗1=−(α3y+α4z+α5t)zrx+(α1x+α4y+α6z+α7t)r+(α2x+α5y+α7z+α8t)s,α∗2=(α2x+α5y+α7z+α8t)x3,α∗3=α3x4+2α5x2(z2+2xy)+α8(z2+2xy)2,α∗4=−(α3x2+α5(z2+2xy))zrx+(α4x2+α7(z2+2xy))r+(α5x2+α8(z2+2xy))s,α∗5=(α5x2+α8(z2+2xy))x3,α∗6=−(α4r−α3zrx+α5s)zrx+(α6r−α4zrx+α7s)r+(α7r−α5zrx+α8s)s−(α3y+α4z+α5t)x2−(α2x+α5y+α7z+α8t)(z2+2xy),α∗7=(α7r−α5zrx+α8s)x3,α∗8=α8x6. $ |
We are interested in
$y=−α22+α2α3+α4α72α27x,z=−α2α7x,s=−3α22α3+α2(α23+6α4α7)+α7(α3α4+2α6α7)4α37√x3,t=α27(α24−2α1α7)+α32α3+α22(α23+3α4α7)+2α2α7(α3α4+α6α7)2α47x,$ |
we have
$y=−α2α5+α27α25x,z=α7α5x,s=α3α7−α4α5α25√x3,t=α2α3α5+α25α6+3α4α5α7−2α3α27α35x,$ |
we have
3.
Summarizing, we have the following distinct orbits:
$⟨∇1+α∇3+∇5⟩O(α)=O(η45α)=O(−η35α)=O(η25α)=O(−η5α),⟨α∇1+∇3+β∇6+∇8⟩O(α,β)=O(−α,β)=O(η3α,η23β)=O(−η3α,η23β)=O(−η23α,−η3β)=O(η23α,−η3β),⟨α∇1+∇4+∇8⟩O(α)=O(−α)=O(η45α)=O(−η45α)=O(η35α)=O(−η35α)=O(η25α)=O(−η25α)=O(η5α)=O(−η5α),⟨∇1+∇8⟩,⟨∇3+∇5⟩,⟨∇3+∇7⟩,⟨∇5⟩,⟨∇6+∇8⟩,⟨∇7⟩,⟨∇8⟩,$ |
which gives the following new algebras:
$ Nα125:e1e1=e2e1e2=e4e1e3=e5e2e2=αe5e2e4=e5e3e3=e4Nα,β126:e1e1=e2e1e2=e4e1e3=αe5e2e2=e5e3e3=e4+βe5e4e4=e5Nα127:e1e1=e2e1e2=e4e1e3=αe5e2e3=e5e3e3=e4e4e4=e5N128:e1e1=e2e1e2=e4e1e3=e5e3e3=e4e4e4=e5N129:e1e1=e2e1e2=e4e2e2=e5e2e4=e5e3e3=e4N130:e1e1=e2e1e2=e4e2e2=e5e3e3=e4e3e4=e5N131:e1e1=e2e1e2=e4e2e4=e5e3e3=e4N132:e1e1=e2e1e2=e4e3e3=e4+e5e4e4=e5N133:e1e1=e2e1e2=e4e3e3=e4e3e4=e5N134:e1e1=e2e1e2=e4e3e3=e4e4e4=e5 $ |
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} & H2D(N4∗11)=⟨[Δ12],[Δ22],[Δ23],[Δ33]⟩H2C(N4∗11)=H2D(N4∗11)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩ & \phi = (x0000x200z0x30t2xzsx4)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ12],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. $ |
Take
$ \phi^T(0α10α2α1α3α4α50α4α6α7α2α5α7α8)\phi = (α∗α∗1α∗∗α∗2α∗1α∗3+α∗∗α∗4α∗5α∗∗α∗4α∗6α∗7α∗2α∗5α∗7α∗8), $ |
we have
$ α∗1=(α1x+α4z+α5t)x2+2(α2x+α7z+α8t)xz,α∗2=(α2x+α7z+α8t)x4,α∗3=(α3x2+4α5xz+4α8z2)x2−(α6z+α7t)x3−(α2x+α7z+α8t)s,α∗4=(α4x+2α7z)x4+(α5x+2α8z)xs,α∗5=(α5x+2α8z)x5,α∗6=α6x6+2α7x3s+α8s2,α∗7=(α7x3+α8s)x4,α∗8=α8x8. $ |
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
$x=α2α5,z=−α22α4+α2α3α54α35,s=−α32α4α45,t=(α2α4+2α22)(α2α4+α3α5)−4α1α2α254α45,$ |
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
$ x=α2α5,z=α2(α2α4+α3α5)α25α6−4α25,s=−α4α32α45,t=α2(2α22α4+α3α4α5+α2(α24+2α3α5)−α1α5(4α5−α6))α35(4α5−α6), $ |
we have the family of representatives
3.
4.
Summarizing, we have the following distinct orbits:
$⟨∇1+∇2−2∇4+α∇6⟩O(α)=O(−α),⟨∇1+α∇3+β∇4+γ∇6+∇8⟩O(α,β,γ)=O(−η5α,η25β,−η35γ)=O(η25α,η45β,−η5γ)=O(−η35α,−η5β,η45γ)=O(η45α,−η35β,η25γ),⟨∇1+α∇4+β∇5+∇7⟩O(α,β)=O(α,−β)=O(−α,−iβ)=O(−α,iβ),⟨α∇2+∇3+∇5+4∇6⟩O(α)=O(−α),⟨∇2+α∇4⟩,⟨∇2+α∇4+∇6⟩,⟨∇2+∇5+α∇6⟩,⟨∇3+α∇4+β∇6+∇8⟩,⟨∇4+α∇5+∇7⟩,⟨∇4+α∇6+∇8⟩,⟨∇5+α∇6⟩,⟨∇5+∇7⟩,⟨∇6+∇8⟩,⟨∇7⟩,⟨∇8⟩,$ |
which gives the following new algebras:
$ Nα135:e1e1=e2e1e2=e5e1e3=e4e1e4=e5e2e2=e4e2e3=−2e5e3e3=αe5Nα,β,γ136:e1e1=e2e1e2=e5e1e3=e4e2e2=e4+αe5e2e3=βe5e3e3=γe5e4e4=e5Nα,β137:e1e1=e2e1e2=e5e1e3=e4e2e2=e4e2e3=αe5e2e4=βe5e3e4=e5Nα138:e1e1=e2e1e3=e4e1e4=αe5e2e2=e4+e5e2e4=e5e3e3=4e5Nα139:e1e1=e2e1e3=e4e1e4=e5e2e2=e4e2e3=αe5Nα140:e1e1=e2e1e3=e4e1e4=e5e2e2=e4e2e3=αe5e3e3=e5Nα141:e1e1=e2e1e3=e4e1e4=e5e2e2=e4e2e4=e5e3e3=αe5Nα,β142:e1e1=e2e1e3=e4e2e2=e4+e5e2e3=αe5e3e3=βe5e4e4=e5Nα143:e1e1=e2e1e3=e4e2e2=e4e2e3=e5e2e4=αe5e3e4=e5Nα144:e1e1=e2e1e3=e4e2e2=e4e2e3=e5e3e3=αe5e4e4=e5Nα145:e1e1=e2e1e3=e4e2e2=e4e2e4=e5e3e3=αe5N146:e1e1=e2e1e3=e4e2e2=e4e2e4=e5e3e4=e5N147:e1e1=e2e1e3=e4e2e2=e4e3e3=e5e4e4=e5N148:e1e1=e2e1e3=e4e2e2=e4e3e4=e5N149:e1e1=e2e1e3=e4e2e2=e4e4e4=e5 $ |
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} & H2D(N4∗12)=⟨[Δ12],[Δ13],[Δ23],[Δ33]⟩H2C(N4∗12)=H2D(N4∗12)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩ & \phi_{\pm} = (x0000x20000±x20t0sx4)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ12],∇2=[Δ13],∇3=[Δ14],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. $ |
Take
$ \phi_{\pm}^T(0α1α2α3α10α4α5α2α4α6α7α3α5α7α8)\phi_{\pm} = (α∗α∗1α∗2α∗3α∗10α∗4α∗5α∗2α∗4α∗6α∗7α∗3α∗5α∗7α∗8), $ |
we have
$ α∗1=(α1x+α5t)x2,α∗2=(α3x+α8t)s±(α2x+α7t)x2,α∗3=(α3x+α8t)x4,α∗4=(α5s±α4x2)x2,α∗5=α5x6,α∗6=α6x4±2α7sx2+α8s2,α∗7=(α8s±α7x2)x4,α∗8=α8x8. $ |
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:
$⟨∇1+α∇2+β∇4+γ∇5+μ∇6+∇8⟩O(α,β,γ,μ)=O(±α,±η45β,η25γ,η45μ)=O(±α,∓η35β,η45γ,−η35μ)=O(±α,±η25β,−η5γ,η25μ)=O(±α,∓η5β,−η35γ,−η5μ),⟨∇1+∇3+α∇4+β∇6⟩O(α,β)=O(−α,β)=O(α,−β)=O(−α,−β),⟨∇1+α∇3+β∇4+∇7⟩O(α,β)=O(−η3α,η23β)=O(η23α,−η3β),⟨∇2+α∇3+∇5+β∇6+γ∇7⟩O(α,β,γ)=O(−α,β,−γ)=O(−η3α,η23β,γ)=O(η3α,η23β,−γ)=O(η23α,−η3β,γ)=O(−η23α,−η3β,−γ),⟨∇2+α∇4+β∇5+γ∇6+∇8⟩O(α,β,γ)=O(−α,β,γ)=O(η45α,η25β,η45γ)=O(−η45α,η25β,η45γ)=O(−η35α,η45β,−η35γ)=O(η35α,η45β,−η35γ)=O(η25α,−η5β,η25γ)=O(−η25α,−η5β,η25γ)=O(−η5α,−η35β,−η5γ)=O(η5α,−η35β,−η5γ),⟨∇3⟩,⟨∇3+∇4+α∇6⟩,⟨∇3+α∇4+∇7⟩O(α)=O(−α),⟨∇3+∇5+α∇6+β∇7⟩O(α,β)=O(α,−β),⟨∇3+∇6⟩,⟨∇4+α∇5+β∇6+∇8⟩O(α,β)=O(−iα,−β)=O(iα,−β)=O(−α,β),⟨∇4+∇7⟩,⟨∇5+∇6+α∇7⟩O(α,β)=O(α,−β),⟨∇5+α∇6+∇8⟩,⟨∇5+α∇7⟩O(α)=O(−α),⟨∇6+∇8⟩,⟨∇7⟩,⟨∇8⟩,$ |
which gives the following new algebras:
$ Nα,β,γ,μ150:e1e1=e2e1e2=e5e1e3=αe5e2e2=e4e2e3=βe5e2e4=γe5e3e3=e4+μe5e4e4=e5Nα,β151:e1e1=e2e1e2=e5e1e4=e5e2e2=e4e2e3=αe5e3e3=e4+βe5Nα,β152:e1e1=e2e1e2=e5e1e4=αe5e2e2=e4e2e3=βe5e3e3=e4e3e4=e5Nα,β,γ153:e1e1=e2e1e3=e5e1e4=αe5e2e2=e4e2e4=e5e3e3=e4+βe5e3e4=γe5Nα,β,γ154:e1e1=e2e1e3=e5e2e2=e4e2e3=αe5e2e4=βe5e3e3=e4+γe5e4e4=e5N155:e1e1=e2e1e4=e5e2e2=e4e3e3=e4Nα156:e1e1=e2e1e4=e5e2e2=e4e2e3=e5e3e3=e4+αe5Nα157:e1e1=e2e1e4=e5e2e2=e4e2e3=αe5e3e3=e4e3e4=e5Nα,β158:e1e1=e2e1e4=e5e2e2=e4e2e4=e5e3e3=e4+αe5e3e4=βe5N159:e1e1=e2e1e4=e5e2e2=e4e3e3=e4+e5Nα,β160:e1e1=e2e2e2=e4e2e3=e5e2e4=αe5e3e3=e4+βe5e4e4=e5N161:e1e1=e2e2e2=e4e2e3=e5e3e3=e4e3e4=e5Nα162:e1e1=e2e2e2=e4e2e4=e5e3e3=e4+e5e3e4=αe5Nα163:e1e1=e2e2e2=e4e2e4=e5e3e3=e4+αe5e4e4=e5Nα164:e1e1=e2e2e2=e4e2e4=e5e3e3=e4e3e4=αe5N165:e1e1=e2e2e2=e4e3e3=e4+e5e4e4=e5N166:e1e1=e2e2e2=e4e3e3=e4e3e4=e5N167:e1e1=e2e2e2=e4e3e3=e4e4e4=e5 $ |
Here we will collect all information about
$ N4∗13(λ)e1e1=e2e1e2=e3e1e3=e4e2e2=λe4H2D(N4∗13(2))=⟨[Δ22],4[Δ23]+[Δ14],[Δ24]⟩,H2C(N4∗13(2))=H2D(N4∗13(2))⊕⟨[Δ23],[Δ33],[Δ34],[Δ44]⟩H2D(N4∗13(λ)λ≠2)=⟨[Δ22],(3λ−2)[Δ23]+[Δ14]⟩,H2C(N4∗13(λ)λ≠2)=H2D(N4∗13(λ)⊕⟨[Δ23],[Δ24],[Δ33],[Δ34],[Δ44]⟩ϕ=(x000yx200z2xyx30tλy2+2xz(λ+2)x2yx4) $ |
Let us use the following notations:
$ ∇1=[Δ14]+(3λ−2)[Δ23],∇2=[Δ22],∇3=[Δ23],∇4=[Δ24],∇5=[Δ33],∇6=[Δ34],∇7=[Δ44]. $ |
Take
$\phi^T(000α10α2(3λ−2)α1+α3α40(3λ−2)α1+α3α5α6α1α4α6α7)\phi = (α∗∗α∗∗∗α∗α∗1α∗∗∗α∗2+λα∗(3λ−2)α∗1+α∗3α∗4α∗(3λ−2)α∗1+α∗3α∗5α∗6α∗1α∗4α∗6α∗7),$ |
we have
$ α∗1=(α1x+α4y+α6z+α7t)x4,α∗2=α2x4+4λ(α6y+α7z)xy2+λ2α7y4+4(α4z+(α3+(3λ−2)α1)y)x3+2(4α6yz+2α7z2+(2α5+λα4)y2)x2−λ((λ+2)(α4y+α6z+α7t)y+((α3+4λα1)y+α5z+α6t)x)x2,α∗3=[(λ+2)(α4x2+2α6xy+2α7xz+λα7y2)y+((α3+(3λ−2)α1)x2+2α5xy+2α6xz+λα6y2)x]x2−(3λ−2)(α1x+α4y+α6z+α7t)x4,α∗4=(α4x2+2α6xy+2α7xz+λα7y2)x4,α∗5=(α5x2+2(λ+2)α6xy+(λ+2)2α7y2)x4,α∗6=(α6x+(λ+2)α7y)x6,α∗7=α7x8. $ |
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
$ x=α3+2(λ−2)α1α4,y=−α1(α3+2(λ−2)α1)α24,z=(2(2−λ)α1−α3)(α2α4+(λ−4)α1α3+(3λ2−12λ+8)α21)4α34,t=0,$ |
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
$x=α3α4−2α1(2α4+α5)α4α5),y=α1(2α1(2α4+α5)−α3α4)α24α5),z=(2α1(2α4+α5)−α3α4)(α2α24−4α1α3α4+4α21(2α4+α5))4α44α5,t=0,$ |
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
$x=√4λ(λ−4)α1α3+λ2α2α5+4(3λ3−12λ2+8λ+16)α21λα5,y=−4α1√4λ(λ−4)α1α3+λ2α2α5+4(3λ3−12λ2+8λ+16)α21λ2α25,z=0,t=0,$ |
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.
$y=−α6α7(λ+2)x,z=2(λ+2)2α4α7−(λ+4)α262(λ+2)2α27x,t=(λ2+6λ+8)α4α6α7−2(λ+2)2α1α27−(λ+4)α362(λ+2)2α37x,$ |
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
$x=α6α7,y=α6(α2α7−α24−2α1α6)6α7(α4α6+8α1α7−α3α7)),z=y2x−α4x2α7−α6yα7,t=−xα1+yα4+zα6α7,$ |
we have the family of representatives
$ \langle \alpha\nabla_3+\nabla_5+\nabla_6+\nabla_7 \rangle_{\alpha\neq0,\lambda = -2}; $ |
$x=α6α7,y=α6(α4α6+8α1α7−α3α7)2α7(−α26+α5α7),z=y2x−α42α7x−α6α7y,t=(α4α6−2α1α7)x2−2α6α7y2+2(α26−α4α7)xy2α27x,$ |
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:
$⟨(λ−4)∇1+4(1−λ)(λ−2)(∇2+∇3)⟩λ∉{1;2;4},⟨∇1+α∇2+∇5⟩λ=0,α≠0,⟨α∇1+∇3⟩α≠0,⟨∇1+∇5⟩,⟨∇2+α∇3+λ4∇4+∇5⟩O(α)=O(−α)λ≠0,⟨∇2+α∇3+β∇5+∇6⟩O(α,β)=O(η23α,−ηβ)=O(−η3α,η23β)λ=0,⟨α∇2+∇3+∇7⟩O(α)=O(−η3α)=O(η23α)λ≠−2,⟨α∇2+β∇5+∇6+∇7⟩λ=−2,⟨α∇2+∇5+∇7⟩,⟨∇2+∇7⟩,⟨∇3+∇4⟩,⟨∇3+α∇4+∇5⟩α≠0,⟨∇3+α∇5+∇6⟩,⟨α∇3+∇5+∇6+∇7⟩α≠0,λ=−2,⟨∇3+∇7⟩λ=−2,⟨∇4⟩λ≠2,⟨α∇4+∇5⟩α≠0,⟨∇5+∇6⟩,⟨∇6⟩,⟨∇7⟩.$ |
Now we have the following new algebras
$ Nλ≠1;2;4168:e1e1=e2e1e2=e3e1e3=e4e1e4=(λ−4)e5e2e2=λe4+4(1−λ)(λ−2)e5e2e3=−λ(λ+2)e5Nα≠0169:e1e1=e2e1e2=e3e1e3=e4e1e4=e5e2e2=αe5e2e3=−2e5e3e3=e5Nλ,α≠0170:e1e1=e2e1e2=e3e1e3=e4e1e4=αe5e2e2=λe4e2e3=(1+α(3λ−2))e5Nλ171:e1e1=e2e1e2=e3e1e3=e4e1e4=e5e2e2=λe4e2e3=(3λ−2)e5e3e3=e5Nλ≠0,α172:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4+e5e2e3=αe5e2e4=λ4e5e3e3=e5Nα,β173:e1e1=e2e1e2=e3e1e3=e4e2e2=e5e2e3=αe5e3e3=βe5e3e4=e5Nλ≠−2,α174:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4+αe5e2e3=e5e4e4=e5Nα,β175:e1e1=e2e1e2=e3e1e3=e4e2e2=−2e4+αe5e3e3=βe5e3e4=e5e4e4=e5Nλ,α176:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4+αe5e3e3=e5e4e4=e5Nλ177:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4+e5e4e4=e5Nλ178:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e2e3=e5e2e4=e5Nλ,α≠0179:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e2e3=e5e2e4=αe5e3e3=e5Nλ,α180:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e2e3=e5e3e3=αe5e3e4=e5Nα≠0181:e1e1=e2e1e2=e3e1e3=e4e2e2=−2e4e2e3=αe5e3e3=e5e3e4=e5e4e4=e5N182:e1e1=e2e1e2=e3e1e3=e4e2e2=−2e4e2e3=e5e4e4=e5Nλ≠2183:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e2e4=e5Nλ,α≠0184:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e2e4=αe5e3e3=e5Nλ185:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e3e3=e5e3e4=e5Nλ186:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e3e4=e5Nλ187:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e4e4=e5 $ |
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} & H2D(N4∗14)=⟨[Δ11],[Δ22],[Δ23],[Δ33]⟩H2C(N4∗14)=H2D(N4∗14)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩ & \phi = (x0000q000rxq0tsxrx2q)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ11],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. $ |
Take
$ \phi^T(α100α20α3α4α50α4α6α7α2α5α7α8)\phi = (α∗1α∗α∗∗α∗2α∗α∗3α∗4α∗5α∗∗α∗4α∗6α∗7α∗2α∗5α∗7α∗8), $ |
we have
$ α∗1=α1x2+2α2xt+α8t2,α∗2=(α2x+α8t)x2q,α∗3=(α3q+α4r+α5s)q+(α4q+α6r+α7s)r+(α5q+α7r+α8s)s,α∗4=(α4q+α6r+α7s)xq+(α5q+α7r+α8s)xr,α∗5=(α5q+α7r+α8s)x2q,α∗6=(α6q2+2α7qr+α8r2)x2,α∗7=(α7q+α8r)x3q,α∗8=α8x4q2. $ |
We are interested in
$x=√α3α6−α24α6,q=α2√α3α6−α24α26,r=−α2α4√α3α6−α24α36,s=0,t=−α1√α3α6−α242α2α6,$ |
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
$x=1,q=√α1α5,r=−α4√α1(α5+α6)√α5,s=(α24(2α5+α6)−α3(α5+α6)2)√α12α5(α5+α6)2√α5,t=0,$ |
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:
$⟨∇1+∇3+α∇4+β∇6+∇8⟩O(α,β)=O(iα,−β)=O(−iα,−β)=O(−α,β),⟨∇1+∇3+α∇6+∇7⟩O(α)=O(−η3α)=O(η23α),⟨∇1+∇4+∇5−∇6⟩,⟨∇1+∇4+α∇6+∇8⟩O(α)=O(−η3α)=O(η23α),⟨∇1+∇5+α∇6⟩,⟨∇1+∇6+∇7⟩,⟨∇1+∇6+∇8⟩,⟨∇1+∇7⟩,⟨∇1+∇8⟩,⟨∇2⟩,⟨∇2+∇3⟩,⟨∇2+∇3+∇6⟩,⟨∇2+∇3+α∇6+∇7⟩O(α)=O(−η3α)=O(η23α),⟨∇2+∇4⟩,⟨∇2+∇4+∇5−∇6⟩,⟨∇2+∇5+α∇6⟩,⟨∇2+∇6⟩,⟨∇2+∇6+∇7⟩,⟨∇2+∇7⟩,⟨∇3+α∇4+β∇6+∇8⟩O(α,β)=O(iα,−β)=O(−iα,−β)=O(−α,β),⟨∇3+α∇6+∇7⟩O(α)=O(−η3α)=O(η23α),⟨∇4+∇5−∇6⟩,⟨∇4+α∇6+∇8⟩O(α)=O(−η3α)=O(η23α),⟨∇5+α∇6⟩,⟨∇6+∇7⟩,⟨∇6+∇8⟩,⟨∇7⟩,⟨∇8⟩,$ |
which gives the following new algebras:
$ Nα,β188:e1e1=e5e1e2=e3e1e3=e4e2e2=e5e2e3=αe5e3e3=βe5e4e4=e5Nα189:e1e1=e5e1e2=e3e1e3=e4e2e2=e5e3e3=αe5e3e4=e5N190:e1e1=e5e1e2=e3e1e3=e4e2e3=e5e2e4=e5e3e3=−e5Nα191:e1e1=e5e1e2=e3e1e3=e4e2e3=e5e3e3=αe5e4e4=e5Nα191:e1e1=e5e1e2=e3e1e3=e4e2e4=e5e3e3=αe5N192:e1e1=e5e1e2=e3e1e3=e4e3e3=e5e3e4=e5N193:e1e1=e5e1e2=e3e1e3=e4e3e3=e5e4e4=e5N194:e1e1=e5e1e2=e3e1e3=e4e3e4=e5N195:e1e1=e5e1e2=e3e1e3=e4e4e4=e5N196:e1e2=e3e1e3=e4e1e4=e5N197:e1e2=e3e1e3=e4e1e4=e5e2e2=e5N198:e1e2=e3e1e3=e4e1e4=e5e2e2=e5e3e3=e5Nα199:e1e2=e3e1e3=e4e1e4=e5e2e2=e5e3e3=αe5e3e4=e5N200:e1e2=e3e1e3=e4e1e4=e5e2e3=e5N201:e1e2=e3e1e3=e4e1e4=e5e2e3=e5e2e4=e5e3e3=−e5Nα202:e1e2=e3e1e3=e4e1e4=e5e2e4=e5e3e3=αe5N203:e1e2=e3e1e3=e4e1e4=e5e3e3=e5N204:e1e2=e3e1e3=e4e1e4=e5e3e3=e5e3e4=e5N205:e1e2=e3e1e3=e4e1e4=e5e3e4=e5Nα,β206:e1e2=e3e1e3=e4e2e2=e5e2e3=αe5e3e3=βe5e4e4=e5Nα207:e1e2=e3e1e3=e4e2e2=e5e3e3=αe5e3e4=e5N208:e1e2=e3e1e3=e4e2e3=e5e2e4=e5e3e3=−e5Nα209:e1e2=e3e1e3=e4e2e3=e5e3e3=αe5e4e4=e5Nα210:e1e2=e3e1e3=e4e2e4=e5e3e3=αe5N211:e1e2=e3e1e3=e4e3e3=e5e3e4=e5N212:e1e2=e3e1e3=e4e3e3=e5e4e4=e5N213:e1e2=e3e1e3=e4e3e4=e5N214:e1e2=e3e1e3=e4e4e4=e5 $ |
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} & H2D(N4∗15)=⟨[Δ11],[Δ22],[Δ23],[Δ33]⟩H2C(N4∗15)=H2D(N4∗15)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩ & \phi = (x0000x2000rx30tsxrx4)\\ \hline \end{array} $ |
Let us use the following notations:
$ ∇1=[Δ11],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. $ |
Take
$ \phi^T(α100α20α3α4α50α4α6α7α2α5α7α8)\phi = (α∗1α∗α∗∗α∗2α∗α∗3+α∗∗α∗4α∗5α∗∗α∗4α∗6α∗7α∗2α∗5α∗7α∗8), $ |
we have
$ α∗1=α1x2+2α2xt+α8t2,α∗2=(α2x+α8t)x4,α∗3=x4α3+2rx2α4+2sx2α5+r2α6+2rsα7+s2α8−x(rxα2+tx2α7+rtα8),α∗4=(α4x2+α6r+α7s)x3+(α5x2+α7r+α8s)xr,α∗5=(α5x2+α7r+α8s)x4,α∗6=(α6x4+2α7x2r+α8r2)x2,α∗7=(α7x2+α8r)x5,α∗8=α8x8. $ |
We are interested in
2.
$x=2α5(α5+α6),s=2α5(α24(2α5+α6)−α3(α5+α6)2),r=−4α4α25(α5+α6),$ |
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
$x=α2α5,r=−α22α4α25(α5+α6),s=α22((α5+α6)(2α24−α2α4)−α3(α5+α6)2−α24α6)2α35(α5+α6)2,t=−α12α5,$ |
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:
$ ⟨∇1+α∇2+β∇6+∇7⟩O(α,β)=O(η25α,−η5β)=O(η45α,η25β)=O(−η5α,−η35β)=O(−η35α,η45β),⟨∇1+α∇3+β∇4+γ∇6+∇8⟩O(α,β,γ)=O(−η3α,β,η23γ)=O(−η3α,−β,η23γ)=O(η23α,−β,−η3γ)=O(η23α,β,−η3γ)=O(α,−β,γ),⟨α∇1+∇4+∇5−∇6⟩,⟨∇1+∇5+α∇6⟩,⟨2∇2+∇3+∇4⟩,⟨∇2+α∇3+∇6⟩,⟨∇2+α∇4⟩,⟨α∇2+∇4+∇5−∇6⟩α≠0,⟨∇2+∇5+α∇6⟩,⟨∇2+α∇6+∇7⟩O(α)=O(−α),⟨∇3+α∇4+β∇6+∇8⟩O(α,β)=O(iα,−β)=O(−iα,−β)=O(−α,β),⟨∇4+α∇6+∇8⟩O(α)=O(−η3α)=O(η23α),⟨∇5+α∇6⟩,⟨∇6+∇7⟩,⟨∇6+∇8⟩,⟨∇7⟩,⟨∇8⟩,$ |
which gives the following new algebras:
$ Nα,β215:e1e1=e5e1e2=e3e1e3=e4e1e4=αe5e2e2=e4e3e3=βe5e3e4=e5Nα,β,γ216:e1e1=e5e1e2=e3e1e3=e4e2e2=e4+αe5e2e3=βe5e3e3=γe5e4e4=e5Nα217:e1e1=αe5e1e2=e3e1e3=e4e2e2=e4e2e3=e5e2e4=e5e3e3=−e5Nα218:e1e1=e5e1e2=e3e1e3=e4e2e2=e4e2e4=e5e3e3=αe5N219:e1e2=e3e1e3=e4e1e4=2e5e2e2=e4+e5e2e3=e5Nα220:e1e2=e3e1e3=e4e1e4=e5e2e2=e4+αe5e3e3=e5Nα221:e1e2=e3e1e3=e4e1e4=e5e2e2=e4e2e3=αe5Nα≠0222:e1e2=e3e1e3=e4e1e4=αe5e2e2=e4e2e3=e5e2e4=e5e3e3=−e5Nα223:e1e2=e3e1e3=e4e1e4=e5e2e2=e4e2e4=e5e3e3=αe5Nα224:e1e2=e3e1e3=e4e1e4=e5e2e2=e4e3e3=αe5e3e4=e5Nα,β225:e1e2=e3e1e3=e4e2e2=e4+e5e2e3=αe5e3e3=βe5e4e4=e5Nα226:e1e2=e3e1e3=e4e2e2=e4e2e3=e5e3e3=αe5e4e4=e5Nα227:e1e2=e3e1e3=e4e2e2=e4e2e4=e5e3e3=αe5N228:e1e2=e3e1e3=e4e2e2=e4e3e3=e5e3e4=e5N229:e1e2=e3e1e3=e4e2e2=e4e3e3=e5e4e4=e5N230:e1e2=e3e1e3=e4e2e2=e4e3e4=e5N231:e1e2=e3e1e3=e4e2e2=e4e4e4=e5 $ |
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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