Gut microbiomes play a role in developing and regulating autoimmune diseases such as multiple sclerosis (MS). We designed this systematic review to summarize the evidence of the effect of gut microbiota in developing pediatric-onset MS.
PubMed, Scopus, EMBASE, Web of Science, Google Scholar, references of the references and conference abstracts were comprehensively searched by two independent researchers. The search was done on January 1st, 2023. Data regarding the total number of patients, the name of the first author, publication year, country of origin, mean age, duration of the disease, body mass index (BMI), type of MS, Expanded Disability Status Scale (EDSS), age at disease onset and stool composition were extracted.
A literature search revealed 4237 published studies. After removing duplicates, we had 2045 records for evaluation. Twenty-three full texts were evaluated, and four case-control studies remained for systematic review. Three studies were conducted in the United States and one in the Netherlands. The number of participants in included studies ranged between 24 and 68. The mean age of patients at the time of study varied between 11.9 and 17.9 years, and the mean age at the onset of the disease ranged between 11.5 and 14.3 years. Most included patients were female. The results show that median richness (the number of unique taxa identified, which was provided by two studies) was higher in controls, and also Margalef index, which was reported by one study was higher in control group than the case group. The results of two studies also demonstrated that median evenness indexes (taxon distribution, Shannon, Simpson) were higher in control groups, as well as PD index (Faith's phylogenic diversity metric).
The result of this systematic review (including four studies) showed disruption of the microbiota-immune balance in pediatric-onset MS cases.
Citation: Sanaz Mehrabani, Mohsen Rastkar, Narges Ebrahimi, Mahsa Ghajarzadeh. Microbiomes and Pediatric onset multiple sclerosis (MS): A systematic review[J]. AIMS Neuroscience, 2023, 10(4): 423-432. doi: 10.3934/Neuroscience.2023031
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Gut microbiomes play a role in developing and regulating autoimmune diseases such as multiple sclerosis (MS). We designed this systematic review to summarize the evidence of the effect of gut microbiota in developing pediatric-onset MS.
PubMed, Scopus, EMBASE, Web of Science, Google Scholar, references of the references and conference abstracts were comprehensively searched by two independent researchers. The search was done on January 1st, 2023. Data regarding the total number of patients, the name of the first author, publication year, country of origin, mean age, duration of the disease, body mass index (BMI), type of MS, Expanded Disability Status Scale (EDSS), age at disease onset and stool composition were extracted.
A literature search revealed 4237 published studies. After removing duplicates, we had 2045 records for evaluation. Twenty-three full texts were evaluated, and four case-control studies remained for systematic review. Three studies were conducted in the United States and one in the Netherlands. The number of participants in included studies ranged between 24 and 68. The mean age of patients at the time of study varied between 11.9 and 17.9 years, and the mean age at the onset of the disease ranged between 11.5 and 14.3 years. Most included patients were female. The results show that median richness (the number of unique taxa identified, which was provided by two studies) was higher in controls, and also Margalef index, which was reported by one study was higher in control group than the case group. The results of two studies also demonstrated that median evenness indexes (taxon distribution, Shannon, Simpson) were higher in control groups, as well as PD index (Faith's phylogenic diversity metric).
The result of this systematic review (including four studies) showed disruption of the microbiota-immune balance in pediatric-onset MS cases.
The algebraic classification (up to isomorphism) of algebras of dimension
The algebraic study of central extensions of associative and non-associative algebras has been an important topic for years (see, for example, [10,20] and references therein). Our method for classifying nilpotent commutative algebras is based on the calculation of central extensions of nilpotent algebras of smaller dimensions from the same variety (first, this method has been developed by Skjelbred and Sund for Lie algebra case in [20]) and the classifications of all complex
Throughout this paper, we use the notations and methods well written in [10], which we have adapted for the commutative case with some modifications. Further in this section we give some important definitions.
Let
Let
ϕθ(x,y)=θ(ϕ(x),ϕ(y)). |
It is easy to verify that
Let
Call the set
The following result shows that every algebra with a non-zero annihilator is a central extension of a smaller-dimensional algebra.
Lemma 1.1. Let
dim(Ann(A))=m≠0. |
Then there exists, up to isomorphism, a unique
Proof. Let
P(xy)=P((x−P(x)+P(x))(y−P(y)+P(y)))=P(P(x)P(y))=[P(x),P(y)]A′. |
Since
Definition 1.2. Let
Our task is to find all central extensions of an algebra
Let
W1=⟨[θ1],[θ2],…,[θs]⟩,W2=⟨[ϑ1],[ϑ2],…,[ϑs]⟩∈Gs(H2(A,C)), |
we easily have that if
Ts(A)={W=⟨[θ1],…,[θs]⟩∈Gs(H2(A,C)):s⋂i=1Ann(θi)∩Ann(A)=0}, |
which is stable under the action of
Now, let
E(A,V)={Aθ:θ(x,y)=s∑i=1θi(x,y)ei and ⟨[θ1],[θ2],…,[θs]⟩∈Ts(A)}. |
We also have the following result, which can be proved as in [10,Lemma 17].
Lemma 1.3. Let
Orb⟨[θ1],[θ2],…,[θs]⟩=Orb⟨[ϑ1],[ϑ2],…,[ϑs]⟩. |
This shows that there exists a one-to-one correspondence between the set of
The idea of the definition of a
((xy)a)b+((xb)a)y+x((yb)a)=((xy)b)a+((xa)b)y+x((ya)b). |
The above described method gives all commutative (
θ(x,y)=θ(y,x), |
θ((xy)a,b)+θ((xb)a,y)+θ(x,(yb)a)=θ((xy)b,a)+θ((xa)b,y)+θ(x,(ya)b). |
for all
Z2D(D,C)={θ∈Z2C(D,C):θ(x,y)=θ(y,x),θ((xy)a,b)+θ((xb)a,y)+θ(x,(yb)a)=θ((xy)b,a)+θ((xa)b,y)+θ(x,(ya)b) for all x,y,a,b∈D}. |
Observe that
Rs(D)={W∈Ts(D):W∈Gs(H2D(D,C))}, |
Us(D)={W∈Ts(D):W∉Gs(H2D(D,C))}. |
Then
Let us introduce the following notations. Let
NΞj—jth5−dimensional family ofcommutative non−CCD−algebras with parametrs Ξ.Nij—jth i−dimensional non−CCD−algebra.Ni∗j—jth i−dimensional CCD−algebra. |
Remark 1. All families of algebras from our final list do not have intersections, but inside some families of algebras there are isomorphic algebras. All isomorphisms between algebras from a certain family of algebras constucted from the representative
Thanks to [8] we have the complete classification of complex
N3∗01,N4∗01:e1e1=e2H2C=H2DN3∗02,N4∗02:e1e1=e2e1e2=e3H2C≠H2DN3∗03,N4∗03:e1e2=e3H2C=H2DN3∗04,N4∗04:e1e1=e2e2e2=e3H2C≠H2DN4∗05:e1e1=e2e1e3=e4H2C=H2DN4∗06:e1e1=e2e3e3=e4H2C=H2DN4∗07:e1e1=e4e2e3=e4H2C=H2DN4∗08:e1e1=e2e1e2=e3e2e2=e4H2C≠H2DN4∗09:e1e1=e2e2e3=e4H2C≠H2DN4∗10:e1e1=e2e1e2=e4e3e3=e4H2C≠H2DN4∗11:e1e1=e2e1e3=e4e2e2=e4H2C≠H2DN4∗12:e1e1=e2e2e2=e4e3e3=e4H2C≠H2DN4∗13(λ):e1e1=e2e1e2=e3e1e3=e4e2e2=λe4H2C≠H2DN4∗14:e1e2=e3e1e3=e4H2C≠H2DN4∗15:e1e2=e3e1e3=e4e2e2=e4H2C≠H2DN4∗16:e1e2=e3e1e3=e4e2e3=e4H2C≠H2DN4∗17:e1e2=e3e3e3=e4H2C≠H2DN4∗18:e1e1=e4e1e2=e3e3e3=e4H2C≠H2DN4∗19:e1e1=e4e1e2=e3e2e2=e4e3e3=e4H2C≠H2DN401:e1e1=e2e1e2=e3e2e3=e4N402:e1e1=e2e1e2=e3e1e3=e4e2e3=e4N403:e1e1=e2e1e2=e3e3e3=e4N404:e1e1=e2e1e2=e3e2e2=e4e3e3=e4N405:e1e1=e2e1e3=e4e2e2=e3N406:e1e1=e2e1e2=e4e1e3=e4e2e2=e3N407:e1e1=e2e2e2=e3e2e3=e4N408:e1e1=e2e1e3=e4e2e2=e3e2e3=e4N409:e1e1=e2e2e2=e3e3e3=e4N410:e1e1=e2e2e2=e3e1e2=e4e3e3=e4N411(λ):e1e1=e2e1e2=λe4e2e2=e3e2e3=e4e3e3=e4 |
Here we will collect all information about
CohomologyAutomorphismsN3∗02e1e1=e2e1e2=e3H2D(N3∗02)=⟨[Δ13],[Δ22]⟩,H2C(N3∗02)=H2D(N3∗02)⊕⟨[Δ23],[Δ33]⟩ϕ=(x00yx20z2xyx3) |
Let us use the following notations:
∇1=[Δ13],∇2=[Δ22],∇3=[Δ23],∇4=[Δ33]. |
Take
ϕT(00α10α2α3α1α3α4)ϕ=(α∗α∗∗α∗1α∗∗α∗2α∗3α∗1α∗3α∗4), |
we have
α∗1=(α1x+α3y+α4z)x3,α∗2=(α2x2+4α3xy+4α4y2)x2,α∗3=(α3x+2α4y)x4,α∗4=α4x6. |
We are interested only in
θ1=α1∇1+α2∇2+α3∇3+α4∇4 and θ2=β1∇1+β2∇2+β3∇3. |
Thus, we have
α∗1=(α1x+α3y+α4z)x3,β∗1=(β1x+β3y)x3,α∗2=(α2x2+4α3xy+4α4y2)x2,β∗2=(β2x+4β3y)x3,α∗3=(α3x+2α4y)x4,β∗3=β3x5.α∗4=α4x6. |
Consider the following cases.
x=2α24β2, y=−α3α4β2, z=α23(−2β1+β2)+2α4(α2β1−α1β2), |
we have the representatives
x=4β3α4,y=−β2α4,z=β2α3−4α1β3, |
we have the representative
x=4β1−β24β3,y=β22−4β1β216β23,z=(4β1−β2)(8β1α3β3−4β1β2α4−8α1β33+β22α4)32β33α4, |
we have the representative
x=√4α2β23−4β2α3β3+β22α44β23α4, y=−β2√α4β22−4α3β2β3+4α2β238β23√α4, z=(8β1α3β3−4β1β2α4−8α1β33+β22α4)√4α2β23−4β2α3β3+β22α416β33α4√α4, |
we have the family of representatives
Summarizing, we have the following distinct orbits:
⟨∇1,∇2+∇4⟩,⟨∇1+4∇2,−24(∇2+∇3)⟩,⟨∇1+λ∇2,∇3⟩,⟨∇1+λ∇2,∇4⟩,⟨α∇1+∇3,∇2+∇4⟩,⟨∇1+∇3,∇4⟩,⟨∇2,−3∇3⟩,⟨∇2,∇4⟩,⟨∇3,∇4⟩. |
Note that the algebras constructed from the orbits
N12:e1e1=e2e1e2=e3e1e3=e4e2e2=e5e3e3=e5N4168:e1e1=e2e1e2=e3e1e3=e4e2e2=4e4−24e5e2e3=−24e5Nλ,0170:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e2e3=e5Nλ,0184:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e3e3=e5Nα13:e1e1=e2e1e2=e3e1e3=αe4e2e2=e5e2e3=e4e3e3=e5N14:e1e1=e2e1e2=e3e1e3=e4e2e3=e4e3e3=e5N−176:e1e1=e2e1e2=e3e2e2=e4e2e3=−3e5N080:e1e1=e2e1e2=e3e2e2=e4e3e3=e5N15:e1e1=e2e1e2=e3e2e3=e4e3e3=e5 |
Here we will collect all information about
N3∗04e1e1=e2e2e2=e3H2D(N3∗04)=⟨[Δ12]⟩,H2C(N3∗04)=H2D(N3∗04)⊕⟨[Δ13],[Δ23],[Δ33]⟩ϕ=(x000x20z0x4) |
Let us use the following notations:
∇1=[Δ12],∇2=[Δ13],∇3=[Δ23],∇4=[Δ33]. |
Take
ϕT(0α1α2α10α3α2α3α4)ϕ=(α∗α∗1α∗2α∗1α∗∗α∗3α∗2α∗3α∗4), |
we have
α∗1=(α1x+α3z)x2,α∗2=(α2x+α4z)x4,α∗3=α3x5,α∗4=α4x8. |
Consider the following cases:
θ1=α1∇1+α2∇2+α3∇3+α4∇4 and θ2=β1∇1+β2∇2+β3∇3. |
Thus, we have
α∗1=(α1x+α3z)x2,β∗1=(β1x+β3z)x2,α∗2=(α2x+α4z)x4,β∗2=β2x5,α∗3=α3x6,β∗3=β3x6.α∗4=α4x8. |
Then we consider the following subcases:
x=√α3α4−1 and z=−α1√α3−1α−14, |
we have the representative
x=√β1β2 and z=(α1β2−β1α2)√β1(β1α4−β2α3)√β2, |
we have the family of representatives
2.
α∗1=(α1x+α3z)x2,β∗1=β1x3,α∗2=α2x5,β∗2=β2x5,α∗3=α3x6, |
and consider the following subcases:
3.
Summarizing, we have the following distinct orbits:
⟨∇1,∇2⟩, ⟨∇1,∇2+∇3⟩, ⟨∇1,∇3⟩, ⟨∇1,∇3+∇4⟩, ⟨∇1,∇4⟩, ⟨∇1+∇2,α∇1+∇3+∇4⟩O(α)=O(−α), ⟨∇1+∇2,∇3⟩, ⟨∇1+∇2,α∇3+∇4⟩α≠1, ⟨β∇1+∇2+∇3,α∇1+∇4⟩, ⟨α∇1+∇3,∇1+∇4⟩O(α)=O(−η3α)=O(η23α), ⟨∇1+∇3,∇4⟩, ⟨∇1+∇4,∇2⟩, ⟨∇2,∇3⟩, ⟨∇2,∇3+∇4⟩, ⟨∇2,∇4⟩, ⟨∇3,∇4⟩. |
Note that, the orbit
N076:e1e1=e2e1e2=e3e1e4=e5e2e2=e4N16:e1e1=e2e1e2=e4e1e3=e5e2e2=e3e2e3=e5N17:e1e1=e2e1e2=e4e2e2=e3e2e3=e5N18:e1e1=e2e1e2=e4e2e2=e3e2e3=e5e3e3=e5N19:e1e1=e2e1e2=e4e2e2=e3e3e3=e5Nα20:e1e1=e2e1e2=e4+αe5e1e3=e4e2e2=e3e2e3=e5e3e3=e5N21:e1e1=e2e1e2=e4e1e3=e4e2e2=e3e2e3=e5Nα≠122:e1e1=e2e1e2=e4e1e3=e4e2e2=e3e2e3=αe5e3e3=e5Nα,β23:e1e1=e2e1e2=βe4+αe5e1e3=e4e2e2=e3e2e3=e4e3e3=e5Nα24:e1e1=e2e1e2=αe4+e5e2e2=e3e2e3=e4e3e3=e5N25:e1e1=e2e1e3=e4e2e2=e3e2e3=e4e3e3=e5N26:e1e1=e2e1e2=e4e1e3=e5e2e2=e3e3e3=e4N27:e1e1=e2e1e3=e4e2e2=e3e2e3=e5N28:e1e1=e2e1e3=e4e2e2=e3e2e3=e5e3e3=e5N29:e1e1=e2e1e3=e4e2e2=e3e3e3=e5N30:e1e1=e2e2e2=e3e2e3=e4e3e3=e5 |
Here we will collect all information about
N4∗02e1e1=e2e1e2=e3H2D(N4∗02)=⟨[Δ13],[Δ22],[Δ14],[Δ24],[Δ44]⟩H2C(N4∗02)=H2D(N4∗02)⊕⟨[Δ23],[Δ33],[Δ34]⟩ϕ=(x000qx200w2xqx3re00t) |
Let us use the following notations:
∇1=[Δ13],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT(00α1α20α3α4α5α1α4α6α7α2α5α7α8)ϕ=(α∗α∗∗α∗1α∗2α∗∗α∗3α∗4α∗5α∗1α∗4α∗6α∗7α∗2α∗5α∗7α∗8), |
we have
α∗1=(α1x+α4q+α6w+α7e)x3,α∗2=(α1x+α4q+α6w+α7e)r+(α2x+α5q+α7w+α8e)t,α∗3=(α3x2+4α4xq+4α6q2)x2,α∗4=(α4x+2α6q)x4,α∗5=(α4r+α5t)x2+2(α6r+α7t)xq,α∗6=α6x6,α∗7=(α6r+α7t)x3,α∗8=α6r2+2α7rt+α8t2. |
We interested in
x=4√α2α4−α1α5,t=α24,r=−α4α5,q=−α14√α2α4−α1α5α4, |
we have the representative
x=α3−4α1α4,t=(α3−4α1)4α24(α2α4−α1α5),r=α5(α3−4α1)4α34(α1α5−α2α4),q=4α21−α1α3α24, |
we have the representative
x=α4α8,t=α34α28,q=−α1α8,r=−α24α5α28,e=α1α5−α2α4, |
we have the representative
x=α3−4α1α4,t=√(α3−4α1)5α24√α8,q=4α21−α1α3α24,r=−α5√(α3−4α1)5α34√α8,e=(4α1−α3)(α2α4−α1α5)α24α8, |
we have the representative
2.
x=2α27,q=−α5α7,e=−2α1α7,w=α25+2α1α8−2α2α7,t=−2α7,r=α8, |
we have the representative
x=1, q=−α52α7, e=−α1α7, w=α25+2α1α8−2α2α72α27, t=α3α7, r=−α3α82α27, |
we have the representative
x=√α7,t=α4,e=α3−4α14√α7,r=−α4α82α7,q=−α3√α74α4,w=4α1α4α8−4α2α4α7+α3(α5α7−α4α8)4α4√α37, |
we have the representative
x=−α32α4+α5α7−α4α82α27,q=α3(α3α27−2α4α5α7+α24α8)8α24α27,w=(α3α27−2α4α5α7+α24α8)(4α2α4α7−4α1α4α8+α3(−α5α7+α4α8))8α24α47,e=(4α1−α3)(α3α27−2α4α5α7+α24α8)8α4α37,t=(α3α27−2α4α5α7+α24α8)24α4α57,r=−α8(α3α27−2α4α5α7+α24α8)28α4α67, |
we have the representative
3.
x=1,t=α26α2α6−α1α7,q=−α42α6,r=α6α7α1α7−α2α6,e=0,w=α24−2α1α6α6, |
we have the representative
x=√α3α6−α24α26,t=√(α3α6−α24)5α36(α2α6−α1α7),q=−α4√α3α6−α242α26,r=−√(α3α6−α24)5α7α46(α2α6−α1α7),e=0, |
and
t=α26α5α6−α4α7x4, q=−α42α6x, r=α6α7α4α7−α5α6x4, e=0, w=α24−2α1α6α6x, |
we have the representatives
t=α6x3√α6α8−α27,q=−α4x2α6,r=−α7x3√α6α8−α27,e=(α1α7−α2α6)xα6α8−α27,w=(α242α26+α1α8−α2α7α27−α6α8)x, |
we have the representatives
x=α5α6−α4α7√α26(α6α8−α27),t=(α5α6−α4α7)3α26(α27−α6α8)2,q=α4(α4α7−α5α6)2α6√α26(α6α8−α27),r=α7(α4α7−α5α6)3α36(α27−α6α8)2,e=α6(α5α6−α4α7)(α4α5α6−α24α7+2α6(−α2α6+α1α7))2α36√(α6α8−α27)3,w=α6(α5α6−α4α7)(α24α8−α4α5α7+2α6(α2α7−α1α8))2α36√(α6α8−α27)3, |
we have the representative
Summarizing, we have the following distinct orbits
⟨∇2+∇3+∇4⟩,⟨∇2+α∇3+∇5+∇6⟩,⟨∇2+∇3+∇6⟩,⟨∇2+∇4⟩,⟨∇2+∇6⟩,⟨∇3+∇4+∇8⟩,⟨∇3+∇5+∇6⟩,⟨α∇3+∇5+∇6+∇8⟩,⟨∇3+∇6+∇8⟩,⟨∇3+∇7⟩,⟨∇4+∇5+∇7⟩,⟨∇4+∇7⟩,⟨∇4+∇8⟩,⟨∇5+∇6⟩,⟨∇6+∇8⟩,⟨∇7⟩, |
which gives the following new algebras:
N31:e1e1=e2e1e2=e3e1e4=e5e2e2=e5e2e3=e5Nα32:e1e1=e2e1e2=e3e1e4=e5e2e2=αe5e2e4=e5e3e3=e5N33:e1e1=e2e1e2=e3e1e4=e5e2e2=e5e3e3=e5N34:e1e1=e2e1e2=e3e1e4=e5e2e3=e5N35:e1e1=e2e1e2=e3e1e4=e5e3e3=e5N36:e1e1=e2e1e2=e3e2e2=e5e2e3=e5e4e4=e5N37:e1e1=e2e1e2=e3e2e2=e5e2e4=e5e3e3=e5Nα38:e1e1=e2e1e2=e3e2e2=αe5e2e4=e5e3e3=e5e4e4=e5N39:e1e1=e2e1e2=e3e2e2=e5e3e3=e5e4e4=e5N40:e1e1=e2e1e2=e3e2e2=e5e3e4=e5N41:e1e1=e2e1e2=e3e2e3=e5e2e4=e5e3e4=e5N42:e1e1=e2e1e2=e3e2e3=e5e3e4=e5N43:e1e1=e2e1e2=e3e2e3=e5e4e4=e5N44:e1e1=e2e1e2=e3e2e4=e5e3e3=e5N45:e1e1=e2e1e2=e3e3e3=e5e4e4=e5N46:e1e1=e2e1e2=e3e3e4=e5 |
Here we will collect all information about
N4∗04e1e1=e2e2e2=e3H2D(N4∗04)=⟨[Δ12],[Δ14],[Δ24],[Δ44]⟩,H2C(N4∗04)=H2D(N4∗04)⊕⟨[Δ13],[Δ23],[Δ33],[Δ34]⟩ϕ=(x0000x200y0x4rz00t) |
Let us use the following notations:
∇1=[Δ12],∇2=[Δ13],∇3=[Δ14],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT(0α1α2α3α10α4α5α2α4α6α7α3α5α7α8)ϕ=(α∗α∗1α∗2α∗3α∗1α∗∗α∗4α∗5α∗2α∗4α∗6α∗7α∗3α∗5α∗7α∗8), |
we have
α∗1=(α1x+α4y+α5z)x2,α∗2=(α2x+α6y+α7z)x4,α∗3=(α2x+α6y+α7z)r+(α3x+α7y+α8z)t,α∗4=α4x6,α∗5=(α4r+α5t)x2,α∗6=α6x8,α∗7=(α6r+α7t)x4,α∗8=α6r2+2α7rt+α8t2. |
We interested in
x=α2α5, t=α42α25, z=−α1α2, r=−α32α3α25, y=0, |
we have the representative
x=α3, t=√α2α28, z=−α3, r=0, y=0, |
we have the representative
x=√α1α2−1, t=4√α51α−32√α−18, z=−√α1α−12α3α−18, r=0, y=0, |
we have the representative
x=α25α2α8,t=α55α22α38,z=−α1α5α2α8,r=α45(α1α8−α3α5)α32α38,y=0, |
we have the representative
2.
3.
4.
(b)
(c)
x=5√α1α−16,t=10√α81α−36α−58,z=−α3α−185√α1α−16,y=0,r=0, |
we have the representative
x=√α4α−16, t=α24√α−36α−18, z=−α3α−18√α4α−16, y=0, r=0, |
we have the representative
Summarizing, we have the following distinct orbits:
⟨∇1+∇2+∇8⟩,⟨∇1+∇3+∇6⟩,⟨α∇1+∇3+∇4+∇6⟩O(α)=O(−α),⟨α∇1+β∇4+∇5+∇6+∇8⟩O(α,β)=O(−α,β)=O(±iα,−β),⟨α∇1+∇4+∇6+∇8⟩O(α)=O(−α),⟨∇1+∇4+∇7⟩,⟨α∇1+∇4+∇5+∇7⟩O(α)=O(−α),⟨∇1+∇5+∇7⟩,⟨∇1+∇6+∇8⟩,⟨∇1+∇7⟩,⟨∇2+∇3+∇4⟩,⟨∇2+∇4+∇8⟩,⟨∇2+∇5⟩,⟨∇2+∇5+∇8⟩,⟨∇2+∇8⟩,⟨∇3+∇4⟩,⟨α∇3+∇4+∇5+∇6⟩O(α)=O(−α),⟨∇3+∇5+∇6⟩,⟨∇3+∇6⟩,⟨∇4+∇7⟩,⟨∇4+∇8⟩,⟨∇5+∇6⟩,⟨∇5+∇7⟩,⟨∇6+∇8⟩,⟨∇7⟩. |
Hence, we have the following new algebras:
N47:e1e1=e2e1e2=e5e1e3=e5e2e2=e3e4e4=e5N48:e1e1=e2e1e2=e5e1e4=e5e2e2=e3e3e3=e5Nα49:e1e1=e2e1e2=αe5e1e4=e5e2e2=e3e2e3=e5e3e3=e5Nα,β50:e1e1=e2e1e2=αe5e2e2=e3e2e3=βe5e2e4=e5e3e3=e5e4e4=e5Nα51:e1e1=e2e1e2=αe5e2e2=e3e2e3=e5e3e3=e5e4e4=e5N52:e1e1=e2e1e2=e5e2e2=e3e2e3=e5e3e4=e5Nα53:e1e1=e2e1e2=αe5e2e2=e3e2e3=e5e2e4=e5e3e4=e5N54:e1e1=e2e1e2=e5e2e2=e3e2e4=e5e3e4=e5N55:e1e1=e2e1e2=e5e2e2=e3e3e3=e5e4e4=e5N56:e1e1=e2e1e2=e5e2e2=e3e3e4=e5N57:e1e1=e2e1e3=e5e1e4=e5e2e2=e3e2e3=e5N58:e1e1=e2e1e3=e5e2e2=e3e2e3=e5e4e4=e5N59:e1e1=e2e1e3=e5e2e2=e3e2e4=e5N60:e1e1=e2e1e3=e5e2e2=e3e2e4=e5e4e4=e5N61:e1e1=e2e1e3=e5e2e2=e3e4e4=e5N62:e1e1=e2e1e4=e5e2e2=e3e2e3=e5Nα63:e1e1=e2e1e4=αe5e2e2=e3e2e3=e5e2e4=e5e3e3=e5N64:e1e1=e2e1e4=e5e2e2=e3e2e4=e5e3e3=e5N65:e1e1=e2e1e4=e5e2e2=e3e3e3=e5N66:e1e1=e2e2e2=e3e2e3=e5e3e4=e5N67:e1e1=e2e2e2=e3e2e3=e5e4e4=e5N68:e1e1=e2e2e2=e3e2e4=e5e3e3=e5N69:e1e1=e2e2e2=e3e2e4=e5e3e4=e5N70:e1e1=e2e2e2=e3e3e3=e5e4e4=e5N71:e1e1=e2e2e2=e3e3e4=e5 |
Here we will collect all information about
N4∗08e1e1=e2e1e2=e3e2e2=e4H2D(N4∗08)=⟨[Δ13],[Δ14]+3[Δ23]⟩H2C(N4∗08)=H2D(N4∗08)⊕⟨[Δ14],[Δ24],[Δ33],[Δ34],[Δ44]⟩ϕ=(x000yx200z2xyx30ty2x2yx4) |
Let us use the following notations:
∇1=[Δ13],∇2=[Δ14]+3[Δ23],∇3=[Δ14],∇4=[Δ24],∇5=[Δ33],∇6=[Δ34],∇7=[Δ44]. |
Take
ϕT(00α1α2+α3003α2α4α13α2α5α6α2+α3α4α6α7)ϕ=(α∗α∗∗α∗1α∗2+α∗3α∗∗α∗∗∗3α∗2α∗4α∗13α∗2α∗5α∗6α∗2+α∗3α∗4α∗6α∗7), |
we have
α∗1=(α1x+3α2y+α5z+α6t)x3+((α2+α3)x+α4y+α6z+α7t)x2y,α∗2=13(3α2x3+(α4+2α5)x2y+3α6xy2+α7y3)x2,α∗3=((α2+α3)x+α4y+α6z+α7t)x4−13(3α2x3+(α4+2α5)x2y+3α6xy2+α7y3)x2,α∗4=(α4x2+2α6xy+α7y2)x4,α∗5=(α5x2+2α6xy+α7y2)x4,α∗6=(α6x+α7y)x6,α∗7=α7x8. |
We are interested in
⟨−14∇2+∇3⟩ and ⟨∇1−14∇2+∇3⟩ |
depending on
2.
3.
x=α2+α3α5, y=3α2α3+3α232α25, z=−(α2+α3)(2α1α5+12α2α3+3α23)4α35, |
we have the representative
x=2(α4−α5),y=3α3,z=0,t=0, |
we have the representative
x=2(α2α5−α2α4+α3α5)+α3α42(α25−α4α5),y=3α3(2(α2α5−α2α4+α3α5)+α3α4)2α5(α5−α4)2,z=−(2α2(α4−α5)−α3(α4+2α5))(4α1(α4−α5)2−24α2α3(α4−α5)+3α23(α4+2α5))8(α4−α5)3α25,t=0, |
we have the family of representatives
4. if
z=y2−α3α6+2y(α5−α4)3α6 and t=−x2α1+xy(4α2+α3)+xzα5+y(yα4+zα6)α6), |
we have
5. if
Summarizing all cases we have the following distinct orbits
⟨∇1−14∇2+∇3⟩,⟨α∇1+∇2+β∇4+∇7⟩O(α,β)=O(−η3α,η23β)=O(η23α,−η3β),⟨∇1+α∇3+∇4⟩O(α)=O(−α),⟨∇1+α∇4+∇7⟩O(α)=O(−α),⟨α∇2+∇3⟩α≠0,−1,⟨∇2+α∇4+∇5⟩,⟨α∇2+β∇4+∇5+∇7⟩O(α,β)=O(−α,β),⟨∇3+∇4+∇5⟩,⟨α∇4+∇5⟩α≠0,1,⟨∇4+α∇5+∇6⟩,⟨∇6⟩, |
which gives the following new algebras:
N72:e1e1=e2e1e2=e3e1e3=e5e1e4=34e5e2e2=e4e2e3=−34e5Nα,β73:e1e1=e2e1e2=e3e1e3=αe5e1e4=e5e2e2=e4e2e3=3e5e2e4=βe5e4e4=e5Nα74:e1e1=e2e1e2=e3e1e3=e5e1e4=αe5e2e2=e4e2e4=e5Nα75:e1e1=e2e1e2=e3e1e3=e5e2e2=e4e2e4=αe5e4e4=e5Nα≠0,−176:e1e1=e2e1e2=e3e1e4=(1+α)e5e2e2=e4e2e3=3αe5Nα77:e1e1=e2e1e2=e3e1e4=e5e2e2=e4e2e3=3e5e2e4=αe5e3e3=e5Nα,β78:e1e1=e2e1e2=e3e1e4=αe5e2e2=e4e2e3=3αe5e2e4=βe5e3e3=e5e4e4=e5N79:e1e1=e2e1e2=e3e1e4=e5e2e2=e4e2e4=e5e3e3=e5Nα≠0,180:e1e1=e2e1e2=e3e2e2=e4e2e4=αe5e3e3=e5Nα81:e1e1=e2e1e2=e3e2e2=e4e2e4=e5e3e3=αe5e3e4=e5N82:e1e1=e2e1e2=e3e2e2=e4e3e4=e5 |
Here we will collect all information about
N4∗09e1e1=e2e2e3=e4H2D(N4∗09)=⟨[Δ12],[Δ13],[Δ22],[Δ33]⟩H2C(N4∗09)=H2D(N4∗09)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩ϕ=(x0000x20000r0t0sx2r) |
Let us use the following notations:
∇1=[Δ12],∇2=[Δ13],∇3=[Δ14],∇4=[Δ22],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT(0α1α2α3α1α40α5α20α6α7α3α5α7α8)ϕ=(α∗α∗1α∗2α∗3α∗1α∗4α∗∗α∗5α∗1α∗∗α∗6α∗7α∗3α∗5α∗7α∗8), |
we have
α∗1=(α1x+α5t)x2,α∗2=(α2x+α7t)r+(α3x+α8t)s,α∗3=(α3x+α8t)x2r,α∗4=α4x4,α∗5=α5x4r,α∗6=(α6r+α7s)r+(α7r+α8s)s,α∗7=(α7r+α8s)x2r,α∗8=α8r2x4. |
We are interested in
x=α1α−14, r=α1α−13, s=−α1α2α−23, t=0, |
we have the family of representatives
2.
x=4√α4α6α−25, r=α4α5−1, t=−α14√α4α6α−65,s=0, |
we have the representative
r=1,x=3√α2α5−1,t=−α13√α2α−45,s=0, |
we have the representative
x=3√α2α5−1, r=α−163√α42α−15, t=−α13√α2α−45,s=0, |
we have the representative
x=3√α2α−15,r=α4α5−1,t=−α13√α2α−45,s=0, |
we have the family of representatives
x=α3α5−1,r=α43α−35α−16,s=−α2α33α−35α−16,t=−α1α3α−25, |
we have the representative
3.
4.
Summarizing, we have the following distinct orbits:
⟨∇1+∇2+α∇4+β∇5+γ∇6+∇8⟩O(α,β,γ)=O(−η5α,−η35β,−η5γ)=O(η25α,−η5β,η25γ)=O(−η35α,η45β,−η35γ)=O(η45α,η25β,η45γ),⟨∇1+∇3⟩,⟨∇1+∇3+α∇4+β∇5+∇7⟩,⟨∇1+∇3+∇4+α∇6⟩,⟨∇1+∇3+∇6⟩,⟨∇1+∇4+α∇5+β∇6+∇8⟩O(α,β)=O(−α,β),⟨∇1+∇4+α∇5+∇7⟩O(α,β)=O(−α,β),⟨∇1+∇5+α∇6+∇8⟩,⟨∇1+∇5+∇7⟩,⟨∇1+∇6+∇8⟩,⟨∇1+∇7⟩,⟨∇1+∇8⟩,⟨∇2+∇4+∇5+α∇6⟩O(α)=O(η3α)=O(η23α),⟨∇2+∇4+α∇5+β∇6+∇8⟩,⟨∇2+∇5⟩,⟨∇2+∇5+∇6⟩,⟨∇2+∇5+α∇6+∇8⟩O(α,β)=O(−α,β)=O(α,η23β)=O(−α,η23β)=O(−α,−η3β)=O(α,−η3β),⟨∇2+∇6+∇8⟩,⟨∇2+∇8⟩,⟨∇3⟩,⟨∇3+∇4⟩,⟨∇3+∇4+∇5+α∇6⟩,⟨∇3+α∇4+∇5+∇7⟩,⟨∇3+∇4+∇6⟩,⟨∇3+∇5⟩,⟨∇3+∇5+∇6⟩,⟨∇3+∇6⟩,⟨∇4+∇5⟩,⟨∇4+∇5+∇6⟩,⟨∇4+α∇5+∇6+∇8⟩O(α)=O(−α),⟨∇4+∇5+∇7⟩,⟨∇4+∇5+∇8⟩,⟨∇4+∇7⟩,⟨∇4+∇8⟩,⟨∇5⟩,⟨∇5+∇6⟩,⟨∇5+∇6+∇8⟩,⟨∇5+∇7⟩,⟨∇5+∇8⟩,⟨∇6+∇8⟩,⟨∇7⟩,⟨∇8⟩, |
which gives the following new algebras:
Nα,β,γ83:e1e1=e2e1e2=e5e1e3=e5e2e2=αe5e2e3=e4e2e4=βe5e3e3=γe5e4e4=e5N84:e1e1=e2e1e2=e5e1e4=e5e2e3=e4Nα,β85:e1e1=e2e1e2=e5e1e4=e5e2e2=αe5e2e3=e4e2e4=βe5e3e4=e5Nα86:e1e1=e2e1e2=e5e1e4=e5e2e2=e5e2e3=e4e3e3=αe5N87:e1e1=e2e1e2=e5e1e4=e5e2e3=e4e3e3=e5Nα,β88:e1e1=e2e1e2=e5e2e2=e5e2e3=e4e2e4=αe5e3e3=βe5e4e4=e5Nα89:e1e1=e2e1e2=e5e2e2=e5e2e3=e4e2e4=αe5e3e4=e5Nα90:e1e1=e2e1e2=e5e2e3=e4e2e4=e5e3e3=αe5e4e4=e5N91:e1e1=e2e1e2=e5e2e3=e4e2e4=e5e3e4=e5N92:e1e1=e2e1e2=e5e2e3=e4e3e3=e5e4e4=e5N93:e1e1=e2e1e2=e5e2e3=e4e3e4=e5N94:e1e1=e2e1e2=e5e2e3=e4e4e4=e5Nα95:e1e1=e2e1e3=e5e2e2=e5e2e3=e4e2e4=e5e3e3=αe5Nα,β96:e1e1=e2e1e3=e5e2e2=e5e2e3=e4e2e4=αe5e3e3=βe5e4e4=e5N97:e1e1=e2e1e3=e5e2e3=e4e2e4=e5N98:e1e1=e2e1e3=e5e2e3=e4e2e4=e5e3e3=e5Nα99:e1e1=e2e1e3=e5e2e3=e4e2e4=e5e3e3=αe5e4e4=e5N100:e1e1=e2e1e3=e5e2e3=e4e3e3=e5e4e4=e5N101:e1e1=e2e1e3=e5e2e3=e4e4e4=e5N102:e1e1=e2e1e4=e5e2e3=e4N103:e1e1=e2e1e4=e5e2e2=e5e2e3=e4Nα104:e1e1=e2e1e4=e5e2e2=e5e2e3=e4e2e4=e5e3e3=αe5Nα105:e1e1=e2e1e4=e5e2e2=αe5e2e3=e4e2e4=e5e3e4=e5N106:e1e1=e2e1e4=e5e2e2=e5e2e3=e4e3e3=e5N107:e1e1=e2e1e4=e5e2e3=e4e2e4=e5N108:e1e1=e2e1e4=e5e2e3=e4e2e4=e5e3e3=e5N109:e1e1=e2e1e4=e5e2e3=e4e3e3=e5N110:e1e1=e2e2e2=e5e2e3=e4e2e4=e5N111:e1e1=e2e2e2=e5e2e3=e4e2e4=e5e3e3=e5Nα112:e1e1=e2e2e2=e5e2e3=e4e2e4=αe5e3e3=e5e4e4=e5N113:e1e1=e2e2e2=e5e2e3=e4e2e4=e5e3e4=e5N114:e1e1=e2e2e2=e5e2e3=e4e2e4=e5e4e4=e5N115:e1e1=e2e2e2=e5e2e3=e4e3e4=e5N116:e1e1=e2e2e2=e5e2e3=e4e4e4=e5N117:e1e1=e2e2e3=e4e2e4=e5N118:e1e1=e2e2e3=e4e2e4=e5e3e3=e5N119:e1e1=e2e2e3=e4e2e4=e5e3e3=e5e4e4=e5N120:e1e1=e2e2e3=e4e2e4=e5e3e4=e5N121:e1e1=e2e2e3=e4e2e4=e5e4e4=e5N122:e1e1=e2e2e3=e4e3e3=e5e4e4=e5N123:e1e1=e2e2e3=e4e3e4=e5N124:e1e1=e2e2e3=e4e4e4=e5 |
Here we will collect all information about
N4∗10e1e1=e2e1e2=e4e3e3=e4H2D(N4∗10)=⟨[Δ13],[Δ14],[Δ22],[Δ23],[Δ33]⟩H2C(N4∗10)=H2D(N4∗10)⊕⟨[Δ24],[Δ34],[Δ44]⟩ϕ=(x000yx2−zrx0z0r0tz2+2xysx3),r2=x3 |
Let us use the following notations:
∇1=[Δ13],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT(00α1α20α3α4α5α1α4α6α7α2α5α7α8)ϕ=(α∗α∗∗α∗1α∗2α∗∗α∗3α∗4α∗5α∗1α∗4α∗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
N4∗11e1e1=e2e1e3=e4e2e2=e4H2D(N4∗11)=⟨[Δ12],[Δ22],[Δ23],[Δ33]⟩H2C(N4∗11)=H2D(N4∗11)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩ϕ=(x0000x200z0x30t2xzsx4) |
Let us use the following notations:
∇1=[Δ12],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT(0α10α2α1α3α4α50α4α6α7α2α5α7α8)ϕ=(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=√α1α2,z=0,s=α1α3√α1α22√α2,t=0, |
we have the family of representatives
x=α4+2α2,z=−α1,s=α3(α4+2α2)3α2,t=0, |
we have the the family of representatives
x=α2α6,z=−α1α2α6(α4+2α2),s=α1α22α6+2α32α3+α22α3α4α36(α4+2α2),t=0, |
we have the family of representatives
2.
x=4α5,z=−α3,s=−64α4α25,t=α3α4−4α1α5α5 |
we have the representative
x=α2α5,z=−α22α4+α2α3α54α35,s=−α32α4α45,t=(α2α4+2α22)(α2α4+α3α5)−4α1α2α254α45, |
we have the representative
x=α2α5,z=0,s=−α32α4α45,t=−α1α2α25, |
we have the representative
x=√α2α4+α3α5α5,z=0, s=−α4√(α2α4+α3α5)3α45, t=−α1√α2α4+α3α5α25, |
we have the family of representatives
x=α6−4α5,z=α3, s=α4(4α5−α6)3α5, t=4α1α5−α1α6−α3α4α5, |
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
N4∗12e1e1=e2e2e2=e4e3e3=e4H2D(N4∗12)=⟨[Δ12],[Δ13],[Δ23],[Δ33]⟩H2C(N4∗12)=H2D(N4∗12)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩ϕ±=(x0000x20000±x20t0sx4) |
Let us use the following notations:
∇1=[Δ12],∇2=[Δ13],∇3=[Δ14],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT±(0α1α2α3α10α4α5α2α4α6α7α3α5α7α8)ϕ±=(α∗α∗1α∗2α∗3α∗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.
⟨∇2+α∇4+β∇5+γ∇6+∇8⟩; |
⟨∇1+α∇2+β∇4+γ∇5+μ∇6+∇8⟩. |
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
ϕT(000α10α2(3λ−2)α1+α3α40(3λ−2)α1+α3α5α6α1α4α6α7)ϕ=(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
(α3,α4,α5,α6,α7)≠(0,0,0,0,0) and (α1,α4,α6,α7)≠(0,0,0,0). |
Let us consider the following cases:
⟨α∇1+∇3⟩α∉{0,(λ−4)4(1−λ)(λ−2)};λ≠1,2,4; |
2.
x=4α24,y=−4α1α4,z=α1α3(4−λ)−α2α4−α21(8−12λ+3λ2),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=α1α5,y=−α1α32α25,z=α1(2α2α5+(λ−2)α23+4(λ2−3λ+2)α1α3)2λα35,t=0, |
we have the family of representatives
(b) if
x=4α34, y=−4α1α24, z=4α1α3α4−α2α24−4α21(2α4+α5), 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=λα3+2(λ2−2λ−4)α1λα5,y=−4α1(λα3+2(λ2−2λ−4)α1)λ2α25,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=−α1α4x,z=−α2α24+(λ−4)α1α3α4+α21(4α5+(3λ2−12λ+8)α4)α24(4α4−λα5)x,t=0, |
we have two families of representatives
⟨α∇4+∇5⟩α≠λ4 and ⟨∇3+α∇4+∇5⟩α≠λ4 |
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=√α5α7,y=8α1−α32√α5α7,z=α7(α3−8α1)2−2α4α254α5α7√α5α7,t=α3α4−2α1(4α4+α5)2α7√α5α7, |
we have the family of representatives
(c) if
x=α6α7,y=0,z=−α4α62α27,t=α6(α4α6−2α1α7)2α37, |
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
⟨α∇3+∇5+∇6+∇7⟩α≠0,λ=−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
⟨α∇2+β∇5+∇6+∇7⟩β≠1,λ=−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
N4∗14e1e2=e3e1e3=e4H2D(N4∗14)=⟨[Δ11],[Δ22],[Δ23],[Δ33]⟩H2C(N4∗14)=H2D(N4∗14)⊕⟨[Δ14],[Δ24],[Δ34],[Δ44]⟩ϕ=(x0000q000rxq0tsxrx2q) |
Let us use the following notations:
∇1=[Δ11],∇2=[Δ14],∇3=[Δ22],∇4=[Δ23],∇5=[Δ24],∇6=[Δ33],∇7=[Δ34],∇8=[Δ44]. |
Take
ϕT(α100α20α3α4α50α4α6α7α2α5α7α8)ϕ=(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=−α4α5q,s=2α24−α3α52α25q,t=0, |
we have the representatives
x=α5,q=α2,r=−α2α4α5,s=α2(2α24−α3α5)2α25,t=−α1α52α2, |
we have the representatives
x=1,q=√α1α5,r=0,s=−α3√α12α5√α5,t=0, |
we have the representative
x=α4α5,q=1,r=0,s=−α32α5,t=0, |
we have the representative
x=α4α5,q=√α1α5,r=0,s=−α3√α12α5√α5,t=0, |
we have the representative
x=α5,q=α2,r=0,s=−α2α32α5,t=−α1α52α2, |
we have the representative
x=α4α5,q=α2α4α25,r=0,s=−α2α3α42α35,t=−α1α42α2α5, |
we have the representative
(d) if
x=1,q=1,s=α24(2α5+α6)−α3(α5+α6)22α5(α5+α6)2,r=−α4α5+α6,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=α5,q=α2, r=−α2α4α5+α6, s=α2(α24(2α5+α6)−α3(α5+α6)2)2α5(α5+α6)2,t=−α1α52α2, |
we have the family of representatives
3.
x=3√α3α7,q=6√α31α3α27,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:
Here we will collect all information about
Let us use the following notations:
Take
we have
We are interested in
2.
we have the family of representatives
we have the family of representatives
we have the family of representatives
(b) if
depending on
3.
we have
4.
Summarizing all cases we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
in the case
and on the opposite case, for
We are interested in
2.
we have the following family of representatives
we have the following family of representatives
3.
we have the family of representatives
we have the family of representatives
4.
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
then in the case
and in the opposite case
We are interested in
2.
3.
we have the family of representatives
we have the family of representatives
4.
we have the family of representatives
we have the family of representatives
we have the family of representatives
we have the family of representatives
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
We are interested in
2.
3.
4.
Summarizing all cases, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
then, in the case
For define the main families of representatives, we will use
Let us consider the following cases:
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
We are interested in
(b) if
2.
(b) if
(c) if
3.
4.
Summarizing all cases, we have the following distinct orbits:
Hence, we have the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
We are interested in
2. if
3. if
we have the family of representatives
4. if
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
We are interested in
2.
3.
4.
we have the family of representatives
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
We are interested in
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
We are interested in
(b)
2.
3.
(b)
4.
we have the family of representatives
we have the family of representatives
we have the family of representatives
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
Since
2. if
3. if
4. if
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
We are interested in
2.
3.
4.
we have family of representatives
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
Since
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
Since
2.
3.
we have the family of representatives
4.
we have the family of representatives
we have the family of representatives
we have the family of representatives
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
Since
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Here we will collect all information about
Let us use the following notations:
Take
we have
Since
Summarizing, we have the following distinct orbits:
which gives the following new algebras:
Remark 2. Note that the algebras
Theorem 5.1. Let
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1. | Patrícia Damas Beites, Amir Fernández Ouaridi, Ivan Kaygorodov, The algebraic and geometric classification of transposed Poisson algebras, 2023, 117, 1578-7303, 10.1007/s13398-022-01385-4 |