G | K1 | Decomposition | No. |
E7 | A1 | m=T⊕A5⊕p1⊕p2⊕p3 | 6 |
A5 | m=T⊕A1⊕p1⊕p2⊕p3 | 4 | |
A1×A5 | m=T⊕p1⊕p2⊕p3 | 2 | |
E6 | A1 | m=T⊕A1⊕A3⊕p1⊕p2⊕p3 | 10 |
A3 | m=T⊕A1⊕A1⊕p1⊕p2⊕p3 | 4 | |
A1×A3 | m=T⊕A1⊕p1⊕p2⊕p3 | 4 | |
A1×A1 | m=T⊕A3⊕p1⊕p2⊕p3 | 2 | |
A1×A1×A3 | m=T⊕p1⊕p2⊕p3 | 1 |
In this article, we find several new non-Riemannian Einstein-Randers metrics on some homogeneous manifolds arising from the generalized Wallach spaces. We first prove the existence of Riemannian Einstein metrics on these homogeneous manifolds. Based on these metrics, we prove that there exist non-Riemannian Einstein-Randers metrics on these homogeneous manifolds.
Citation: Xiaosheng Li. New Einstein-Randers metrics on certain homogeneous manifolds arising from the generalized Wallach spaces[J]. AIMS Mathematics, 2023, 8(10): 23062-23086. doi: 10.3934/math.20231174
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In this article, we find several new non-Riemannian Einstein-Randers metrics on some homogeneous manifolds arising from the generalized Wallach spaces. We first prove the existence of Riemannian Einstein metrics on these homogeneous manifolds. Based on these metrics, we prove that there exist non-Riemannian Einstein-Randers metrics on these homogeneous manifolds.
Randers metrics were introduced by Randers in the context of general relativity, and later named after him by Ingarden. A Randers metric F is built from a Riemannian metric and a 1-form, i.e.,
F=α+β |
where α is a Riemannian metric and β is a 1-form and the length of β corresponding to the Riemannian metric α is less than 1 everywhere.
Then, F is Riemannian if and only if F(x,y)=F(x,−y). But sometimes it is convenient to use the following definition of a Randers metric in [2], i.e.,
F(x,y)=√[h(W,y)]2+h(y,y)λλ−h(W,y)λ. | (1.1) |
Here, λ=1−h(W,W)>0, and F is Riemannian if and only if W=0. The pair (h,W) is called the navigation data of the corresponding Randers metric F.
The Ricci scalar Ric(x,y) of a Finsler metric is defined by the sum of those n−1 flag curvatures K(x,y,ev), where {ev:v=1,2,⋯,n−1} is any collection of n−1 orthonormal transverse edges perpendicular to the flagpole, i.e.,
Ric(x,y)=n−1∑v=1Rvv. | (1.2) |
The Ricci tensor is defined by
Ricij=(12F2Ric)yiyj. | (1.3) |
The Ricci scalar depends on the position x and the flagpole y (see [2,3]), but does not depend on the specific n−1 flags with transverse edges orthogonal to y. It is well known that the Ricci scalar in Riemannian geometry only depends on x. Thus, it is quite interesting to study a Finsler manifold whose Ricci scalar does not depend on the flagpole y. Generally, a Finsler metric with such a property is called an Einstein metric, i.e.,
Ric(x,y)=(n−1)K(x) | (1.4) |
for some function K(x) on M. In particular, for a Randers manifold (M,F) with dimM≥3, F is an Einstein metric if and only if K(x) is a constant on M (see [2]). The following lemma is an important result on Einstein-Randers metrics.
Lemma 1.1 ([2]). Suppose that (M,F) is a Randers space with the navigation data (h,W). Then, (M,F) is Einstein with Ricci scalar Ric(x)=(n−1)K(x) if and only if there exists a constant σ satisfying the following conditions:
1) h is Einstein with Ricci scalar (n−1)K(x)+116σ2, and
2) W is an infinitesimal homothety of h, i.e., LWh=−σh.
Furthermore, σ must be zero whenever h is not Ricci-flat.
It is well known that K(x) is a constant if (M,F) is a homogeneous Einstein Finsler manifold. Here, a Finsler manifold (M,F) is called homogeneous if its full group of isometry acts transitively on M. Based on Lemma 1.1, Deng-Hou obtained a characterization of homogeneous Einstein-Randers metrics.
Lemma 1.2 ([8]). Let G be a connected Lie group and H a closed subgroup of G such that G/H is a reductive homogeneous space with a decomposition g=h+m. Suppose that h is a G-invariant Riemannian metric on G/H and W∈m is invariant under H with h(W,W)<1. Let ˜W be the corresponding G-invariant vector field on G/H with ˜W|o=W. Then, the Randers metric F with the navigation data (h,˜W) is Einstein with the Ricci constant K if and only if h is Einstein with the Ricci constant K and W satisfies
⟨[W,X]m,Y⟩+⟨X,[W,Y]m⟩=0,∀X,Y∈m, | (1.5) |
where ⟨,⟩ is the restriction of h on To(G/H)≃m. In this case, ˜W is a Killing vector field with respect to the Riemannian metric h.
Just as in the Riemannian case, it is a fundamental problem to classify homogeneous Einstein Finsler spaces. In particular, it is very important to know if a homogeneous manifold admits invariant Einstein Finsler metrics. In fact, there are many studies on homogenous Einstein Randers manifolds [5,6,9,10,11,14,15,16,17] and a little progress on homogeneous Einstein (α,β)-metrics [18].
The main goal of this paper is to find non-Riemannian Einstein-Randers metrics on homogeneous manifolds G/H for G=E7 and G=E6, which means we prove the following theorem.
Theorem 1.3.
1) There are at least six families of G-invariant non-Riemannian Einstein-Randers metrics on the homogeneous manifold E7/A1.
2) There are at least four families of G-invariant non-Riemannian Einstein-Randers metrics on the homogeneous manifold E7/A5.
3) There are at least two families of G-invariant non-Riemannian Einstein-Randers metrics on the homogeneous manifold E7/(A1×A5).
4) There are at least ten families of G-invariant non-Riemannian Einstein-Randers metrics on the homogeneous manifold E6/A1.
5) There are at least four families of G-invariant non-Riemannian Einstein-Randers metrics on the homogeneous manifold E6/A3.
6) There are at least four families of G-invariant non-Riemannian Einstein-Randers metrics on the homogeneous manifold E6/(A1×A3).
7) There are at least two families of G-invariant non-Riemannian Einstein-Randers metrics on the homogeneous manifold E6/(A1×A1).
8) There is at least one family of G-invariant non-Riemannian Einstein-Randers metric on the homogeneous manifold E6/(A1×A1×A3).
Let G be a compact semi-simple Lie group with Lie algebra g, K a connected closed subgroup of G with Lie algebra k. Through this paper, we denote by B the negative of the Killing form of g, which is positive definite because of the compactness of G. As a result, B can be treated as an inner product on g. Let g=k⊕m be the reductive decomposition with respect to B such that [k,m]⊂m, where m is the tangent space of G/K. We assume that m can be decomposed into mutually non-equivalent irreducible Ad(K)-modules as follows:
m=m1⊕⋯⊕mq. |
We will write k=k0⊕k1⊕⋯⊕kp, where k0=Z(k) is the center of k and ki is the simple ideal for i=1,⋯,p. Let G×K act on G by (g1,g2)g=g1gg−12. Then, G×K acts almost effectively on G with isotropy group Δ(K)={(k,k)|k∈K}. As a result, G can be treated as the coset space (G×K)/Δ(K) and we have g⊕k=Δ(k)⊕Ω, where Ω≅T0((G×K)/Δ(K))≅g via the linear map (X,Y)→(X−Y)∈g, (X,Y)∈Ω.
A direct conclusion is that there exists a 1-1 corresponding between all G-invariant metrics on the reductive homogeneous space G/K and AdG(K)-invariant inner products on m. Now, we have a orthogonal decomposition of g with respect to the Killing form of g: g=k0⊕k1⊕⋯⊕kp⊕m1⊕⋯⊕mq=(k0⊕k1⊕⋯⊕kp)⊕(kp+1⊕⋯⊕kp+q), with m1⊕⋯⊕mq=kp+1⊕⋯⊕kp+q. In addition, we assume that dimRk0≤1 and the ideals ki are mutually non-isomorphic for i=1,⋯,p. Then, we consider all Ad(K)-invariant metrics on G of the form:
⟨,⟩=x0⋅B|k0+x1⋅B|k1+⋯+xp+q⋅B|kp+q, | (2.1) |
where xi∈R+ for i=0,1,⋯,p+q. And all Ad(K)-invariant metrics on G/K of the form:
(,)=xp+1⋅B|kp+1+⋯+xp+q⋅B|kp+q, | (2.2) |
where xi∈R+ for i=p+1,⋯,p+q.
Set from now on di=dimRki and {eiα}diα=1 be a B-orthonormal basis adapted to the decomposition of g which means eiα∈ki and α is the number of basis in ki. Then, we consider the numbers Aγα,β=B([eiα,ejβ],ekγ) such that [eiα,ejβ]=∑γAγα,βekγ, and set
(ijk):=[ij k]=∑(Aγα,β)2, |
where the sum is taken over all indices α,β,γ with eiα∈ki,ejβ∈kj,ekγ∈kk. Then, (ijk) is independent of the choice for the B-orthonormal basis of ki,kj,kk, and symmetric for all three indices which means (ijk)=(jik)=(jki).
In [1] and [13], the authors obtained the formulas for the components of the Ricci tensor with respect to the left-invariant metric given by (2.1), which can be described by the following lemma.
Lemma 2.1. Let G be a compact connected semi-simple Lie group endowed with the left-invariant metric ⟨,⟩ given by (2.1). Then, the components r0,r1,⋯,rp+q of the Ricci tensor associated to ⟨,⟩ are expressed as follows:
rk=12xk+14dk∑j,ixkxjxi[kj i]−12dk∑j,ixjxkxi[jk i],(k=0,1,⋯,p+q). |
Here, the sums are taken over all i=0,1,⋯,p+q. In particular, for each k it holds that
p+q∑i,j[jk i]=p+q∑ij(kij)=dk. |
For k=p+1,⋯,p+q and by considering the sums appearing in the expression of rk only for i,j with p+1≤i,j≤p+q, one obtains the components ˆrp+1,⋯,ˆrp+q of the Ricci tensor ˆr of the G-invariant metric (,) on G/K defined by (2.2).
We recall the definition of generalized Wallach spaces. Let G/K be a reductive homogeneous space, where G is a semi-simple compact connected Lie group, K is a connected closed subgroup of G, g and k are the corresponding Lie algebras, respectively. If m, the tangent space of G/K at o=π(e), can be decomposed into three ad(k)-invariant irreducible summands pairwise orthogonal with respect to B as:
m=m1⊕m2⊕m3, |
satisfying [mi,mi]∈k for i∈{1,2,3} and [mi,mj]⊂mk for {i,j,k}={1,2,3}, then we call G/K a generalized Wallach space.
In 2014, classification for generalized Wallach spaces arising from a compact simple Lie group has been obtained by Nikonorov [12] and Chen, Kang and Liang [7], in particular, Nikonorov investigated the semi-simple case and gave the classification in [12]. For convenience, in this article, we use the notations in [7].
According to [7], each kind of the classification corresponds to two commutative involutive automorphisms of g, the Lie algebra of the compact simple Lie group G. In [4], the authors calculated all the coefficients (ijk) in the expression for the components of Ricci tensor with respect to the metric of form (2.1). In [7], E6-Ⅱ and E7-Ⅱ have the following decomposition in the view of Lie algebra:
E6=T⊕A11⊕A21⊕A3⊕p1⊕p2⊕p3,E7=T⊕A1⊕A5⊕p1⊕p2⊕p3. |
By the notations in the above section, we give the following two lemmas in [4] for later use.
Lemma 3.1. In the case of E7-Ⅱ, the possible non-zero coefficients in the expression for the components of Ricci tensor with respect to the metric of the form (2.1) are as follows:
(033)=49,(044)=59,(055)=0,(345)=203,(111)=13,(133)=1,(144)=0,(155)=53,(222)=353,(233)=359,(244)=709,(255)=353. |
Lemma 3.2. In the case of E6-Ⅱ, the possible non-zero coefficients in the expression for the components of Ricci tensor with respect to the metric of the form (2.1) are as follows:
(044)=1/2,(055)=1/2,(066)=0,(456)=4,(111)=1/2,(144)=0,(155)=1,(166)=3/2,(222)=1/2,(244)=1,(255)=0,(266)=3/2,(333)=5,(344)=5/2,(355)=5/2,(366)=5. |
In this section, we will consider the Einstein metrics on some homogeneous manifolds which can be obtained from the generalized Wallach spaces, namely E7/A1,E7/A5,E7/(A1×A5),E6/A1,E6/A3,E6/(A1×A3),E6/(A1×A1) and E6/(A1×A1×A3).
Case of E7/A1. Consider the homogeneous manifold E7/A1 with the decomposition
m=T⊕A5⊕p1⊕p2⊕p3, | (4.1) |
and Ad(A1)-invariant metrics which is also Ad(T⊕A1⊕A5)-invariant on E7/A1 defined by
< , >=u0⋅B|T+x1⋅B|A5+x2⋅B|p1+x3⋅B|p2+x4⋅B|p3, | (4.2) |
where u0,x1,x2,x3,x4∈R+. By Lemmas 2.1 and 3.1, the components of Ricci tensor with respect to the metric (4.2) are as follows:
{rT=14(4u09x22+5u09x23),rA5=1140(353x1+35x19x22+70x19x23+35x13x24),rp1=12x2+536(x2x3x4−x3x4x2−x4x2x3)−148(4u09x22+35x19x22),rp2=12x3+19(x3x4x2−x4x2x3−x2x3x4)−160(5u09x23+70x19x23),rp3=12x4+112(x4x2x3−x2x3x4−x3x4x2)−18035x13x24. |
We consider the homogeneous Einstein equation as follows:
{rT−rA5=0,rA5−rp1=0,rp1−rp2=0,rp2−rp3=0} |
and it turns out to be equivalent to the following system of equations (we normalize the metric by setting u0=1).
{g0=−3x12x22x32−2x12x22x42−x12x32x42−3x22x32x42+5x1x22x42+4x1x32x42=0,g1=36x12x22x32+24x12x22x42+47x12x32x42−60x1x23x3x4+60x1x2x33x4−216x1x2x32x42+60x1x2x3x43+36x22x32x42+4x1x32x42=0,g2=56x1x22x4−35x1x32x4+108x23x3−216x22x3x4−108x2x33+216x2x32x4−12x2x3x42+4x22x4−4x32x4=0,g3=63x1x2x32−56x1x2x42−12x22x3x4−216x2x32x4+216x2x3x42+84x33x4−84x3x43−4x2x42=0. |
We consider the polynomial ring R=Q[z,x1,x2,x3,x4] and an ideal I, generated by polynomials {g0,g1,g2,g3,zx1x2x3x4−1}. We take a lexicographic ordering > with z>x1>x2>x3>x4 for a monomial ordering on R. Then, with the aid of computer software MAPLE 15, we use the Gröbner basis for the ideal I to contain a polynomial of x4 given by (20x42−72x4+49)⋅h(x4), where h(x4) is a polynomial of degree 30 given by
h(x4)=147263121465141201705316711269370210741339344914420477996120017678192287817728000x430−1102576650209541754792174768351559525030817374433243715583773603796209924867276800x429+4392608446633715641642277193442303841356600810515312912956157487998587768223467520x428−16381030403148760413861847066334150627671304859530037939042221945751621483816924672x427+44632927369562126482344261311667463916531381076564814811273737776429261123742947776x426−105145519130201191374375907381751124532181350157935176774829297302755654321119817568x425+215087055560353068666721180840479776113190447174940731329033435251254151873127864288x424−344518374028048111070412269673269777788246179258239908939790910388711063175367161784x423+445879403133315681874085042923629294832116921460904234436654259616839387520348170596x422−454003150730712855457526821861027903580572913313888888785553759520405925035444447080x421+308171762729585306636518011606003014325716616667221520922199158892726288206642411702x420−88928520907166973360693993136042781786034406240792078393993953506206913911288935648x419−40034430463545533917890194187357087475825295381916586084753660204634626955000430908x418+44518815422796823726900600056989666146339579404372825641721394763313021826539183294x417−13231125163887960569832230144624600324399606190969212487314466195084966981574676247x416−797439538169345403753233795452611566212570589018415600726606800661093425142878880x415+1511979559539474891852062236262989481312808910795807182871248144865160620831183632x414−379199106951137596458809929302879785793133656913117463400584039552474682406503552x413+37797140139775487751864398109781462786788335556812190904364736929666448564191712x412−1785948980557995395564396777431636788073937513566295662291234226016378838797312x411+361185586014234569587833784293818501106527299574429579442258565335789303734400x410−29814163618672798385666919644831689684183641996050223223894494904315006447104x49+5073664811180784912883065274644440790802333871137796008412526195301608704x48+55049909220845662880254271319404494752671319946031114132616313964384415744x47+3753669713205988245353403631212112410702925290650176001229552989542586368x46−461850039539407240955273834215202137394483687360075268757733061237596160x45+14855590404155632214609581796329294787394060775182341751503634575953920x44−151882382457983307126403603474158205457944387452892184268979018137600x43+995700410381405891358618695307405787784707293649196430275379200000x42−13417336677067637864115045789201884045646775900717868981944320000x4+568400491719774288687807984496995610757827345058316786073600000. |
Solving h(x4)=0 numerically, we see that there exist four positive solutions, which are given approximately by x4≈0.4025066142,x4≈1.032699036,x4≈1.438454385,x4≈4.052215439. Further, we remark that x1,x2 and x3 can be expressed by polynomials of x4, thus the corresponding solutions of the system {g0=0,g1=0,g2=0,g3=0,h(x4)=0} with x1x2x3x4≠0 are as follows:
{x1≈0.4492704820,x2≈0.7040380158,x3≈0.7236008036,x4≈0.4025066142},{x1≈0.2625192922,x2≈1.149055866,x3≈0.6923507482,x4≈1.032699036},{x1≈0.3393694928,x2≈0.7028774603,x3≈1.499308774,x4≈1.438454385},{x1≈1.476432886,x2≈1.076444066,x3≈4.066468725,x4≈4.052215439}. |
For 20x42−72x4+49=0, we have x4≈0.9111805582 and x4≈2.688819441, whose corresponding solutions of the system {g0=0,g1=0,g2=0,g3=0,20x42−72x4+49=0} with x1x2x3x4≠0 are as follows:
{x1=x3=1,x2=x4≈0.9111805582},{x1=x3=1,x2=x4≈2.688819441}. |
In conclusion, we find six different G-invariant Einstein metrics on homogeneous manifold E7/A1.
Case of E7/A5. Consider the homogeneous manifold E7/A5 with the decomposition
m=T⊕A1⊕p1⊕p2⊕p3, | (4.3) |
and Ad(A5)-invariant metrics which is also Ad(T⊕A1⊕A5)-invariant on E7/A5 defined by
< , >=u0⋅B|T+x1⋅B|A1+x2⋅B|p1+x3⋅B|p2+x4⋅B|p3, | (4.4) |
where u0,x1,x2,x3,x4∈R+. By Lemmas 2.1 and 3.1, the components of Ricci tensor with respect to the metric (4.4) are as follows:
{rT=14(4u09x22+5u09x23),rA1=112(13x1+x1x22+5x13x24),rp1=12x2+536(x2x3x4−x3x4x2−x4x2x3)−148(4u09x22+x1x22),rp2=12x3+19(x3x4x2−x4x2x3−x2x3x4)−1605u09x23,rp3=12x4+112(x4x2x3−x2x3x4−x3x4x2)−1805x13x24. |
Moreover, the homogeneous Einstein equation is given by
{rT−rA1=0,rA1−rp1=0,rp1−rp2=0,rp2−rp3=0}, |
which is equivalent to the following system of equations by setting u0=1:
{g0=−5x12x22x32−3x12x32x42−x22x32x42+5x1x22x42+4x1x32x42=0,g1=60x12x22x3+45x12x3x42−60x1x23x4+60x1x2x32x4−216x1x2x3x42+60x1x2x43+12x22x3x42+4x1x3x42=0,g2=−9x1x32x4+108x23x3−216x22x3x4−108x2x33+216x2x32x4−12x2x3x42+4x22x4−4x32x4=0,g3=9x1x2x32−12x22x3x4−216x2x32x4+216x2x3x42+84x33x4−84x3x43−4x2x42=0. |
We consider the polynomial ring R=Q[z,x1,x2,x3,x4] and an ideal I, generated by polynomials {g0,g1,g2,g3,zx1x2x3x4−1}. We take a lexicographic ordering > with z>x1>x2>x3>x4 for a monomial ordering on R. Then, with the aid of computer software MAPLE 15, we use the Gröbner basis for the ideal I to contain a polynomial h(x4) of degree 24 given by
h(x4)=743926266335759400800157293103178343319505630645046435156250000x424−4378144578543039011361358385061081569161674400675383078316875000x423+11004903226880209703956063057559301456591525835888793871369802500x422−15728343701831201252751181874054831911519725141563733372843754800x421+15016244958117620330302995693661960097380760512161595698974470890x420−11994308781926707001101666261632390461497787947208181720205033200x419+9738633984043261058161384571494874932415716842246643813641906732x418−6775088500386513521204014691159118392435727285614732468339912598x417+2706243877713720652207013644363017005932182610630242895498110351x416−12000153858619368362412920925916507581760380187119703324718112x415−470852895612592200583477610033508459972546973038296225090613648x414+158657369099271326137588879121559258044753269615019628589825152x413−4309426841561170090469256176045397413256930487964567974681056x412−6417570511115254453560753629391603456981811382551849470944256x411+1526468649104808736601756484180315699890352358674705978019200x410−181418867068729727750349886730055784794192965200077070182912x49+13350908794995611352389703910041292729770053075493309180672x48−632996143303600372111228864933836702374869844580196724736x47+18258540029433839516883828010654743829211229647212972032x46−229558007445965630066366475684636804625527678725234688x45−2954197894924039316661797095646508811861752372301824x44+121322059872735831667919068167865971682917554847744x43+543967044145299748038471088930322206082300116992x42−69016429739436519614913257303724864151171891200x4+849606259908726681491324537499135318226370560. |
By solving h(x4)=0 numerically, we get four different positive solutions, which are given approximately by x4≈0.6449885281,x4≈0.7443959787,x4≈1.060416256,x4≈1.501487029. Further, we remark that x1,x2,x3 can be expressed by x4. Therefore, the real solutions of the system {g0=0,g1=0,g2=0,g3=0,h(x4)=0} with x1x2x3x4≠0 are as follows:
{x1≈1.173286641,x2≈0.9076818373,x3≈0.5898460162,x4≈0.6449885281},{x1≈0.05521141092,x2≈0.9874672655,x3≈0.5836473658,x4≈0.7443959787},{x1≈0.07379520877,x2≈0.6077356344,x3≈1.170652643,x4≈1.060416256},{x1≈1.140144619,x2≈0.6881276139,x3≈1.513843081,x4≈1.501487029}. |
To conclude, we find four different G-invariant Einstein metrics on homogeneous manifold E7/A5.
Case of E7/(A1×A5). Consider the homogeneous manifold E7/(A1×A5) with the decomposition
m=T⊕p1⊕p2⊕p3, | (4.5) |
and Ad(A1×A5)-invariant metrics which is also Ad(T⊕A1⊕A5)-invariant on E7/(A1×A5) defined by
< , >=u0⋅B|T+x1⋅B|p1+x2⋅B|p2+x3⋅B|p3, | (4.6) |
where u0,x1,x2,x3∈R+. By Lemmas 2.1 and 3.1, the components of Ricci tensor with respect to the metric (4.6) are as follows:
{rT=14(4u09x21+5u09x22),rp1=12x1+536(x1x2x3−x2x3x1−x3x1x2)−1484u09x21,rp2=12x2+19(x2x3x1−x3x1x2−x1x2x3)−1605u09x22,rp3=12x3+112(x3x1x2−x1x2x3−x2x3x1). |
The homogeneous Einstein equation is equivalent to the following system of equations (we normalize the metric by setting u0=1):
{g0=−15x13x2+15x1x23−54x1x22x3+15x1x2x32+15x12x3+13x22x3=0,g1=27x13x2−54x12x2x3−27x1x23+54x1x22x3−3x1x2x32+x12x3−x22x3=0,g2=−3x12x2−54x1x22+54x1x2x3+21x23−21x2x32−x1x3=0. |
We consider the polynomial ring R=Q[z,x1,x2,x3] and an ideal I, generated by polynomials {g0,g1,g2,zx1x2x3−1}. We take a lexicographic ordering > with z>x1>x2>x3 for a monomial ordering on R. Then, with the aid of computer software MAPLE 15, we use the Gröbner basis for the ideal I to contain a polynomial h(x3) of degree 8 given by
h(x3)=126500890017116160000x38−443159701524243843840x37+626866608280069521864x36−439964490275093593980x35+146470811201043075009x34−18042589809134073603x33+1095420294616803246x32−33677044349391072x3+438386090662912. |
By solving h(x3)=0 numerically, there are two positive solutions, which are given approximately by x3≈0.7485101873 and x3≈1.034311056. Moreover, we remark that x1 and x2 can be expressed by x3. As a result, the real solutions of the system of equations {g0=0,g1=0,g2=0,h(x3)=0} are
{x1≈0.9908972484,x2≈0.5833878061,x3≈0.7485101873},{x1≈0.6019040936,x2≈1.154991982,x3≈1.034311056}. |
In conclusion, we find two different G-invariant Einstein metrics on the homogeneous manifold E7/(A1×A5).
Case of E6/A1. Consider the homogeneous manifold E6/A11 with the decomposition
m=T⊕A21⊕A3⊕p1⊕p2⊕p3, | (4.7) |
and Ad(A11)-invariant metrics which is also Ad(T⊕A11⊕A21⊕A3)-invariant on E6/A11 defined by
< , >=u0⋅B|T+x1⋅B|A21+x2⋅B|A3+x3⋅B|p1+x4⋅B|p2+x5⋅B|p3, | (4.8) |
where u0,x1,x2,x3,x4,x5∈R+. By Lemmas 2.1 and 3.2, the components of Ricci tensor with respect to the metric (4.8) are as follows:
{rT=14(u02x23+u02x24),rA21=112(12x1+x1x23+3x12x25),rA3=160(5x2+5x22x23+5x22x24+5x2x25),rp1=12x3+18(x3x4x5−x4x5x3−x5x3x4)−132(u02x23+x1x23+5x22x23),rp2=12x4+18(x4x5x3−x5x3x4−x3x4x5)−132(u02x24+5x22x24),rp3=12x5+112(x5x3x4−x3x4x5−x4x5x3)−148(3x12x25+5x2x25). |
We study the following homogeneous Einstein equation
{rT−rA21=0,rA21−rA3=0,rA3−rp1=0,rp1−rp2=0,rp2−rp3=0}, |
which is equivalent to the following system of equations by setting u0=1:
{g0=−3x12x32x42−2x12x42x52−x32x42x52+3x1x32x52+3x1x42x52=0,g1=3x12x2x32x42+2x12x2x42x52−2x1x22x32x42−x1x22x32x52−x1x22x42x52−2x1x32x42x52+x2x32x42x52=0,g2=6x1x2x42x52+16x22x32x42+8x22x32x52+23x22x42x52−24x2x33x4x5+24x2x3x43x5−96x2x3x42x52+24x2x3x4x53+16x32x42x52+3x2x42x52=0,g3=−2x1x42x5+5x2x32x5−5x2x42x5+16x33x4−32x32x4x5−16x3x43+32x3x42x5+x32x5−x42x5=0,g4=6x1x3x42+20x2x3x42−15x2x3x52−8x32x4x5−96x3x42x5+96x3x4x52+40x43x5−40x4x53−3x3x52=0. |
We consider the polynomial ring R=Q[z,x1,x2,x3,x4,x5] and an ideal I, generated by polynomials {g0,g1,g2,g3,g4,zx1x2x3x4x5−1}. We take a lexicographic ordering > with z>x1>x2>x3>x4>x5 for a monomial ordering on R. Then, with the aid of computer software MAPLE 15, we use the Gröbner basis for the ideal I to contain a polynomial (4x52−16x5+11)⋅h(x5), where h(x5) is a polynomial of degree 74. Since the length of the polynomial h(x5) may affect the readers read, we put it in Appendix.
By solving h(x5)=0, there exist eight different positive solutions, which are given approximately by x5≈0.4678383774,x5≈0.8564644408,x5≈1.195992694,x5≈1.331593277,x5≈2.323059800, x5≈2.650131702,x5≈3.078478984,x5≈3.248554868. Moreover, we remark that x1,x2,x3,x4 can be written into polynomials of x5. As a result, we get all solutions of the system {g0=0,g1=0,g2=0,g3=0,g4=0,h(x5)=0} with x1x2x3x4x5≠0, which are given below:
{x1≈0.5079142344,x2≈0.5564563075,x3≈0.7674520218,x4≈0.7344343505,x5≈0.4678383774},{x1≈1.391791299,x2≈0.2653010181,x3≈1.024348994,x4≈0.6953615240,x5≈0.8564644408},{x1≈0.1314059305,x2≈0.2920839602,x3≈1.311645073,x4≈0.6903614938,x5≈1.195992694},{x1≈0.1535432666,x2≈0.3214000505,x3≈0.7137178306,x4≈1.420816234,x5≈1.331593277},{x1≈3.641968638,x2≈1.362098346,x3≈2.353687875,x4≈0.9881377660,x5≈2.323059800},{x1≈1.083726408,x2≈0.7345275471,x3≈0.9316800862,x4≈2.652053038,x5≈2.650131702},{x1≈0.3244500133,x2≈1.435139205,x3≈3.101828021,x4≈1.010069757,x5≈3.078478984},{x1≈0.4746512742,x2≈1.288027846,x3≈1.023041915,x4≈3.259446111,x5≈3.248554868}. |
For 4x52−16x5+11=0, we have two different positive solutions given approximately by x5≈0.8819660112 and x5≈3.118033988. Thus, the solutions of the system of equations {g0=0,g1=0,g2=0,g3=0,g4=0,4x52−16x5+11=0} with x1x2x3x4x5≠0 are as follows:
{x1=x2=x3=1,x4=x5≈0.8819660112},{x1=x2=x3=1,x4=x5≈3.118033988}. |
In conclusion, we find ten different G-invariant Einstein metrics on homogeneous manifold E6/A1.
Remark 4.1. One can consider the homogeneous manifold E6/A21, but the Einstein metrics on which is the same as the above up to isometry.
Case of E6/A3. Consider the homogeneous manifold E6/A3 with the decomposition
m=T⊕A11⊕A21⊕p1⊕p2⊕p3, | (4.9) |
and Ad(A3)-invariant metrics which is also Ad(T⊕A11⊕A21⊕A3)-invariant on E6/A3 defined by
<,>=u0⋅B|T+x1⋅B|A11+x2⋅B|A21+x3⋅B|p1+x4⋅B|p2+x5⋅B|p3, | (4.10) |
where u0,x1,x2,x3,x4,x5∈R+. By Lemmas 2.1 and 3.2, the components of Ricci tensor with respect to the metric (4.10) are as follows:
{rT=14(u02x23+u02x24),rA11=112(12x1+x1x24+3x12x25),rA21=112(12x2+x2x23+3x22x25),rp1=12x3+18(x3x4x5−x4x5x3−x5x3x4)−132(u02x23+x2x23),rp2=12x4+18(x4x5x3−x5x3x4−x3x4x5)−132(u02x24+x1x24),rp3=12x5+112(x5x3x4−x3x4x5−x4x5x3)−148(3x12x25+3x22x25). |
Further, the homogeneous Einstein equation is
{rT−rA11=0,rA11−rA21=0,rA21−rp1=0,rp1−rp2=0,rp2−rp3=0}, |
which is equivalent to the following system of equations by setting u0=1:
{g0=−3x12x32x42−2x12x32x52−x32x42x52+3x1x32x52+3x1x42x52=0,g1=3x12x2x32x42+2x12x2x32x52−3x1x22x32x42−2x1x22x42x52−x1x32x42x52+x2x32x42x52=0,g2=24x22x32x4+22x22x4x52−24x2x33x5+24x2x3x42x5−96x2x3x4x52+24x2x3x53+8x32x4x52+3x2x4x52=0,g3=2x1x32x5−2x2x42x5+16x33x4−32x32x4x5−16x3x43+32x3x42x5+x32x5−x42x5=0,g4=6x1x3x42−6x1x3x52+6x2x3x42−8x32x4x5−96x3x42x5+96x3x4x52+40x43x5−40x4x53−3x3x52=0. |
We consider the polynomial ring R=Q[z,x1,x2,x3,x4,x5] and an ideal I, generated by polynomials {g0,g1,g2,g3,g4,zx1x2x3x4x5−1}. We take a lexicographic ordering > with z>x1>x2>x3>x4>x5 for a monomial ordering on R. Then, with the aid of computer software MAPLE 15, we use the Gröbner basis for the ideal I to contain a polynomial h(x5) of degree 31 given by
h(x5)=1261108165616422801344202981595019832319020116358877329036935168000x531−2151788432405784864427357292265755888622754231825843818424015257600x530−2774119470594771842109777615792789059591610394561214417063934689280x529+5270344831923001751708303816759088216591749544446664977785609519104x528−267759863524378249157490847814955848518971246169904017730950922240x527+3382539429618210066981097554617654004425041655004361517303374807040x526−9157476976548260444114456636484244081952190409782190142886449250304x525+3265956979395568431463257163604494142186261853585696957494114123776x524+1113697025312385709566702871382152926625644868347096267275321556992x523+2112333421951376168597459647362124180457453099511088975109229199360x522−1631290981926813934052954101478705657981981503958339738878928916480x521−1395050115461884401753191844047758010380519323029179088958100982784x520+863710166438884070851640864653697900991831475370745010492288656384x519+241554806478645820779502711256075967575496587804599790265232872960x518−132746306295681893821384103002486474701425792860693217422446582528x517−3267602765928478150920615601297457606838015845305743614683411008x516+8892989132102003650248145850686458075368692869017300184897905088x515−1328201011191036292414037060140630816274620551629810587376210400x514−98351649199565867328202679801373812742043192868226187056696624x513+57195416139228700916611387885001217660977148332951265434464548x512−8584293427926321735849799193107683389353503523382291011168608x511+658987123591863795218834332702519196588690183339615796158964x510−21514437280227836839597312688258324479163856784281755956140x59−562803146421214665893608417569171948144549496211000931459x58+69880133370536372146966764193862531901488645811504750528x57−642571168743645462407828141773462190939783200613014680x56−141173037619723104390974578987052013640153011100310880x55+4922149886224089807286830168142173254755229906374200x54+92795089337275364876653655416844512848233792832000x53−6084529180331760636310164863691628563332526660000x52−51292165983013771433079466375050322032293400000x5+4587282600584194356057367005912663107945250000. |
By solving h(x5)=0 numerically, we have four positive and one negative solutions, which can be given approximately by x5≈0.7601951682,x5≈0.9753459906,x5≈1.141009718, x5≈1.523722394,x5≈−1.755049707. With aid of computer we get all the solutions of system of equations {g0=0,g1=0,g2=0,g3=0,g4=0,h(x5)=0} with x1x2x3x4x5≠0 are as follows:
{x1≈0.1044773890,x2≈1.350603440,x3≈0.9684903881,x4≈0.6352925758,x5≈0.7601951682},{x1≈1.350603440,x2≈0.1044773890,x3≈0.6352925758,x4≈0.9684903881,x5≈0.7601951682},{x1≈0.1100292802,x2≈0.1051859906,x3≈1.141500322,x4≈0.6242772748,x5≈0.9753459906},{x1≈0.1051859906,x2≈0.1100292802,x3≈0.6242772748,x4≈1.141500322,x5≈0.9753459906},{x1≈1.126287414,x2≈1.862577839,x3≈1.181080704,x4≈0.7489214139,x5≈1.141009718},{x1≈1.862577839,x2≈1.126287414,x3≈0.7489214139,x4≈1.181080704,x5≈1.141009718},{x1≈1.191985212,x2≈0.1562720393,x3≈1.546458511,x4≈0.7400787612,x5≈1.523722394},{x1≈0.1562720393,x2≈1.191985212,x3≈0.7400787612,x4≈1.546458511,x5≈1.523722394}. |
We remark that among these solutions, the ones with x5 equal induce the same metrics up to isometry. As a result, there are four different G-invariant Einstein metrics on homogeneous manifold E6/A3.
Case of E6/(A1×A3). Consider the homogeneous manifold E6/(A11×A3) with the decomposition
m=T⊕A21⊕p1⊕p2⊕p3, | (4.11) |
and Ad(A11×A3)-invariant metrics which is also Ad(T⊕A11⊕A21⊕A3)-invariant on E6/(A11×A3) defined by
<,>=u0⋅B|T+x1⋅B|A21+x2⋅B|p1+x3⋅B|p2+x4⋅B|p3, | (4.12) |
where u0,x1,x2,x3,x4∈R+. By Lemmas 2.1 and 3.2, the components of Ricci tensor with respect to the metric (4.12) are as follows:
{rT=14(u02x22+u02x23),rA21=112(12x1+x1x22+3x12x24),rp1=12x2+18(x2x3x4−x3x4x2−x4x2x3)−132(u02x22+x1x22),rp2=12x3+18(x3x4x2−x4x2x3−x2x3x4)−132u02x23,rp3=12x4+112(x4x2x3−x2x3x4−x3x4x2)−1483x12x24. |
Moreover, the homogeneous Einstein equation is
{rT−rA21=0,rA21−rp1=0,rp1−rp2=0,rp2−rp3=0}, |
which is equivalent to the following system of equations by setting u0=1:
{g0=−3x12x22x32−2x12x32x42−x22x32x42+3x1x22x42+3x1x32x42=0,g1=24x12x22x3+22x12x3x42−24x1x23x4+24x1x2x32x4−96x1x2x3x42+24x1x2x43+8x22x3x42+3x1x3x42=0,g2=−2x1x32x4+16x23x3−32x22x3x4−16x2x33+32x2x32x4+x22x4−x32x4=0,g3=6x1x2x32−8x22x3x4−96x2x32x4+96x2x3x42+40x33x4−40x3x43−3x2x42=0. |
We consider the polynomial ring R=Q[z,x1,x2,x3,x4] and an ideal I, generated by polynomials {g0,g1,g2,g3,zx1x2x3x4−1}. We take a lexicographic ordering > with z>x1>x2>x3>x4 for a monomial ordering on R. Then, with the aid of computer software MAPLE 15, we use the Gröbner basis for the ideal I to contain a polynomial h(x4) of degree 24 given by
h(x4)=724760062707181194419174600514595111417722211978444800000x424−4549506617169858886572945697876772369451209831712030720000x423+12407995627074284240831723931001141829288068512693092352000x422−19700816036968099156069552630998662850171899956562598297600x421+21336801748378149912816519240285552502805547195911991459840x420−18881224039130808168709393497734018965438973813620910063616x419+15807728714235458621829548954153222096357328191223379263488x418−11339342221372302981951544924786699818394336801815208656896x417+5112518772844735173801750222596331052099974758094486477824x416−403396001117807932930095402077870112549798991179251580928x415−903609223572125644190343758413680417593338129263631754880x414+461550807018117050045055022022782250907255388226385775616x413−65891887023479142966230863046964022312825636044010288900x412−13675428883180345095301338226487838153532965536692318272x411+7404379070352221664927980007974819413355565861485929092x410−1483997367866112378227770061868493553550293718403156128x49+175268024365864286996306905494287487314022126666412671x48−12983233756373427437475671647726918973459199705907872x47+557098428728836786107911315677130125808601844881240x46−7614872308049576020573960694434807958692132824768x45−444217606400638301563490761267824391591116546648x44+16454149288639736436126283769710330401564494976x43+517849751430653664520517887668936015103631904x42−41649452199426082400520496110816668096290560x4+743025277331758306683208760331078413656560. |
By solving h(x4)=0 numerically, we have four positive solutions, which can be given approximately by x4≈0.7380550950,x4≈0.9342575736,x4≈0.9881689149 and x4≈1.544605922. We remark that x1,x2,x3 can be written into polynomials of x4. As a result, we obtain the solutions of the system of equations {g0=0,g1=0,g2=0,g3=0,h(x4)=0} with x1x2x3x4≠0 as follows:
{x1≈1.330255613,x2≈0.9639605550,x3≈0.6236782194,x4≈0.7380550950},{x1≈0.1012164462,x2≈1.121509362,x3≈0.6122385007,x4≈0.9342575736},{x1≈0.1102891330,x2≈0.6235977518,x3≈1.152510591,x4≈0.9881689149},{x1≈1.194390595,x2≈0.7396962365,x3≈1.566827089,x4≈1.544605922}. |
In conclusion, we find four different G-invariant Einstein metrics on homogeneous manifold E6/(A1×A3).
Remark 4.2. One can consider Einstein metrics on E6/(A21×A3), which is the same as the above results up to isometry.
Case of E6/(A1×A1). Consider the homogeneous manifold E6/(A11×A21) with the decomposition
m=T⊕A3⊕p1⊕p2⊕p3, | (4.13) |
and Ad(A11×A21)-invariant metrics which is also Ad(T⊕A11⊕A21⊕A3)-invariant on E6/(A11×A21) defined by
<,>=u0⋅B|T+x1⋅B|A3+x2⋅B|p1+x3⋅B|p2+x4⋅B|p3, | (4.14) |
where u0,x1,x2,x3,x4∈R+. By Lemmas 2.1 and 3.2, the components of Ricci tensor with respect to the metric (4.14) are as follows:
{rT=14(u02x22+u02x23),rA3=160(5x1+5x12x22+5x12x23+5x1x24),rp1=12x2+18(x2x3x4−x3x4x2−x4x2x3)−132(u02x22+5x12x22),rp2=12x3+18(x3x4x2−x4x2x3−x2x3x4)−132(u02x23+5x12x23),rp3=12x4+112(x4x2x3−x2x3x4−x3x4x2)−1485x1x24. |
Moreover, the homogeneous Einstein equation is
{rT−rA3=0,rA3−rp1=0,rp1−rp2=0,rp2−rp3=0}, |
which is equivalent to the following system of equations by setting u0=1:
{g0=−2x12x22x32−x12x22x42−x12x32x42−2x22x32x42+3x1x22x42+3x1x32x42=0,g1=16x12x22x32+8x12x22x42+23x12x32x42−24x1x23x3x4+24x1x2x33x4−96x1x2x32x42+24x1x2x3x43+16x22x32x42+3x1x32x42=0,g2=5x1x22x4−5x1x32x4+16x23x3−32x22x3x4−16x2x33+32x2x32x4+x22x4−x32x4=0,g3=20x1x2x32−15x1x2x42−8x22x3x4−96x2x32x4+96x2x3x42+40x33x4−40x3x43−3x2x42=0. |
We consider the polynomial ring R=Q[z,x1,x2,x3,x4] and an ideal I, generated by polynomials {g0,g1,g2,g3,zx1x2x3x4−1}. We take a lexicographic ordering > with z>x1>x2>x3>x4 for a monomial ordering on R. Then, with the aid of computer software MAPLE 15, we use the Gröbner basis for the ideal I to contain a polynomial h(x4) of degree 17 given by
h(x4)=110023745349636492444277742390625x417−633649771052061357670508068621875x416+1693219483961674637778856732365000x415−4316833335694592632675706273589750x414+8843142281870361254769815434556400x413−13636914380079309535418557047150830x412+17268089388308771236690983095499516x411−16542332326263296024484260888236418x410+9539252746514719228613145099164214x49−1216363759832722486352208197180776x48−2038138365512054114906312824023776x47+1194675635182956573455161439625750x46−183551938256104631960037855371472x45−20803930135660271330029319947786x44+6531836600529448404797287589404x43+7919351072786394774692666914x42−64535844819989369786392515511x4+3728157545978269069973049171. |
By solving h(x4)=0 numerically, we have two positive and one negative solutions, which can be given approximately by x4≈1.216390732,x4≈3.124923887 and x4≈−0.4548317660. With the aid of a computer, we obtain the solutions of the system of equations {g0=0,g1=0,g2=0,g3=0,h(x4)=0} with x1x2x3x4≠0 as follows:
{x1≈0.2932652315,x2≈1.330112486,x3≈0.6899374515,x4≈1.216390732},{x1≈0.2932652315,x2≈0.6899374515,x3≈1.330112486,x4≈1.216390732},{x1≈1.438513860,x2≈3.147955135,x3≈1.011105585,x4≈3.124923887},{x1≈1.438513860,x2≈1.011105585,x3≈3.147955135,x4≈3.124923887}. |
It is easy to see that the first two solutions induce the same metric up to isometry, and so do the later two solutions. In conclusion, we find two different G-invariant Einstein metrics on homogeneous manifold E6/(A1×A1).
Case of E6/(A1×A1×A3). Consider the homogeneous manifold E6/(A1×A1×A3) with the decomposition
m=T⊕p1⊕p2⊕p3, | (4.15) |
and Ad(A1×A1×A3)-invariant metrics which is also Ad(T⊕A1⊕A1⊕A3)-invariant on E6/(A1×A1×A3) defined by
< , >=u0⋅B|T+x1⋅B|p1+x2⋅B|p2+x3⋅B|p3, | (4.16) |
where u0,x1,x2,x3∈R+. By Lemmas 2.1 and 3.2, the components of Ricci tensor with respect to the metric (4.16) are as follows:
{rT=14(u02x21+u02x22),rp1=12x1+18(x1x2x3−x2x3x1−x3x1x2)−132u02x21,rp2=12x2+18(x2x3x1−x3x1x2−x1x2x3)−132u02x22,rp3=12x3+112(x3x1x2−x1x2x3−x2x3x1). |
The homogeneous Einstein equation is
{rT−rp1=0,rp1−rp2=0,rp2−rp3=0}, |
which is equivalent to the following system of equations by setting u0=1:
{g0=−8x13x2+8x1x23−32x1x22x3+8x1x2x32+8x12x3+9x22x3=0,g1=16x13x2−32x12x2x3−16x1x23+32x1x22x3+x12x3−x22x3=0,g2=−8x12x2−96x1x22+96x1x2x3+40x23−40x2x32−3x1x3=0. |
We consider the polynomial ring R=Q[z,x1,x2,x3] and an ideal I, generated by polynomials {g0,g1,g2,zx1x2x3−1}. We take a lexicographic ordering > with z>x1>x2>x3 for a monomial ordering on R. Then, with the aid of computer software MAPLE 15, we use the Gröbner basis for the ideal I to contain a polynomial h(x3) of degree 5 given by
h(x3)=2359296000x35−6217728000x34+6426104320x33−2832337088x32+316436504x3−12778713. |
By solving h(x3)=0 numerically, we have only one solution, which can be given approximately by x3≈0.9460130230. With the aid of a computer, we obtain the solutions of the system of equations {g0=0,g1=0,g2=0,h(x3)=0} with x1x2x3≠0 as follows:
{x1≈1.131543385,x2≈0.6115481021,x3≈0.9460130230},{x1≈0.6115481021,x2≈1.131543385,x3≈0.9460130230}. |
It is easy to see that these two solutions induce the same metric up to isometry. In conclusion, we find one G-invariant Einstein metric on homogeneous manifold E6/(A1×A1×A3).
To summarize the above conclusions, we write the results in the table below:
G | K1 | Decomposition | No. |
E7 | A1 | m=T⊕A5⊕p1⊕p2⊕p3 | 6 |
A5 | m=T⊕A1⊕p1⊕p2⊕p3 | 4 | |
A1×A5 | m=T⊕p1⊕p2⊕p3 | 2 | |
E6 | A1 | m=T⊕A1⊕A3⊕p1⊕p2⊕p3 | 10 |
A3 | m=T⊕A1⊕A1⊕p1⊕p2⊕p3 | 4 | |
A1×A3 | m=T⊕A1⊕p1⊕p2⊕p3 | 4 | |
A1×A1 | m=T⊕A3⊕p1⊕p2⊕p3 | 2 | |
A1×A1×A3 | m=T⊕p1⊕p2⊕p3 | 1 |
Consider Ad(K)-invariant Einstein metrics on homogenous spaces G/K1 given in the above section. By the equivalence of the adjoint representation and the isotropy representation of K1 on m, the vector field
˜W|gK=d(τ(g))|o(W),∀g∈G,W∈T |
is well-defined, and it is G-invariant (see [2]). Furthermore, one can easily verify the equation
⟨[W,X]m,Y⟩+⟨X,[W,Y]m⟩=0 |
holds for any W∈T and X,Y∈m, using the facts that k0⊂k and the metric is Ad(K)-invariant. Then, by Lemma 1.2 the homogeneous metric
F(x,y)=√[⟨W,y⟩]2+⟨y,y⟩λλ−⟨W,y⟩λ | (5.1) |
is a G-invariant Einstein-Randers metric on G/K1 when ⟨W,W⟩<1, and F is Riemannian if and only if W=0. We proved Theorem 1.3.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was partially supported by NSFC Tianyuan Visiting Scholar Program in Mathematics (No. 12226339), The Key Research Project of Henan Higher Education Institutions (China) (No. 22A110021) and Nanhu Scholars Program for Young Scholars of XYNU. The author would like to thank Prof. Shaoqiang Deng for his helpful discussions.
The author declares no conflict of interest.
h(x5)=28673579100848574881047069574381084050781898649580724543209211321664729662678456497786169904210920939212951086389775541876760318678910211558903308615680000000000x574+279670236049102351139904666053663705638247826260026575536269919664362819973890762700706616107922095061304064066123259096681790256520091895967896792203264000000000x573+1291875369012442867211801075880067073381597480213557796617426583241242991214981592367762230991657789990452362121863422117755960294218661159638345628988211200000000x572−4983801537702228227925113026474382151720503216092876104910840001391075434202484157530847326670319544091020993901548182918350343487734457120686888512158433280000000x571+17216521539856304098571693296595022260329753415833178877106770896256620873221854427717781655120802190456608654524330568510755680052854967978691662703182741504000000x570−47134906903643531043642680121757801481345664708082106420933953814393721136987797798131127670603138786568349683221488745877751131762149626365994897585193825075200000x569+116934384679320454309225137495212457425159336561312377838378290690228344729489573155785414789854881376586557619295688061793323497128548662965095067567582525521920000x568−264392771507680828916864397643085926644639623160755866399798071947508938847235949523041390219916943660436339948477102597136164022858426066663900967401044588888064000x567+496770721553663707507376554776199218983032048404871375438377077297921459444422386661996698177374552393068153603786173411816173304359568897591106702610425943005593600x566−876881536358166894240901459194279748931911750922647333848413653721634575821067743560251000955857738838089131633694758391503095357870203791549339956838253084603842560x565+1353996391599382735671148619891202772655823606889067093509028978601542176282751608583473339088348526857141341056905085655107752360025105043068189219915919928372232192x564−1679108547475437185660568212994645164643772461725891123158580966363405293001006874370650732383155195645634456790427758748319890859958076575934933122883870804729135104x563+2024377183626606775678510053831418719122245342425676478204228441602122708192528859546345061829970886879234512221697201074948559569821991449141717206562113613539049472x562−1643266925862135091118830990819637801462544715335474944082855405559362175761251892455585186736185234872856223681752840248061703226588747598615321276681754310451658752x561+735543913080983684548638476702036533242499827690365272464363154755782790549745208808736666922972021135302433355823980747406533440909093909000356081813697305665077248x560−297791290377203296261214349908028990551101553039219372633701366731253478580889525082759150165161055422721934862929291790810936331477893446142941155917287639786455040x559−950025399861139248442891965419593987991259727120577273513681749959840433597209971153601429658295145115074236743486315152756903019807916795950137388253917216617005056x558−208736051313171339580776808799742219258300209889846233049189615508113925804802328655335137117314201609467888045473168246895879745840848534376859628982488734569070592x557+3578200385136776015176714051045851670854136672526263077653764725264013248628470034447957133886168794767763453768539229230529321482266896205869758729238704226207531008x556−6631585299604318536569718938783023960642216085351443307310092519044513720085352805548995393223770950543435498550016050980151816502058416666705061851236622042973339648x555+12924253038516712848174358815723129761463817540157262076657523471152131635977685421145252473007630427851468881994429248774543593665149693022263863172649649982524096512x554−17242946439332115390144257154763126197566498479064698262283158057699713793157831326879997715988295086656018828311434429036334743236130880952569726910644887583385976832x553+16532707636346024910345102997582872740650882598754317132236965208664482668311245057207235103633788794144223957309151402544007412263682604541971778050717414585237766144x552−15764132248919722297479176089413963928716740552404999511220431172851436913174720954916843485790636175432998993984485631792964821883295906387428243023437708866212593664x551+11151571877421656266869847620148573379209920330713335599040955117048910212055068535862848059978340517275400488881462154823226999430198671154426690003806953657725616128x550−2512841271050755700149051067367972282016567293040229541457812367105034999513712725964520152272357906688465357014242266914953459316250316689749388328150148744883470336x549−500677076164051058426443385400134248987615816137846857500109109415745747151204297709666521875803117629403380212254167517987363160583621714473963960185791024061284352x548+1631016388758565467512592518859426839657469900168933752962407763102017309569815840691732979089318100730964083587827975322580407871028907445757675651679983015712260096x547−4213708435514870879416155735925453727725626413806548303786635356973340409719456058058011958765061919291023427768521167821591536500852703732781275448529779371677417472x546+1238341090380899346712944721372469300695230600317142508447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G | K1 | Decomposition | No. |
E7 | A1 | m=T⊕A5⊕p1⊕p2⊕p3 | 6 |
A5 | m=T⊕A1⊕p1⊕p2⊕p3 | 4 | |
A1×A5 | m=T⊕p1⊕p2⊕p3 | 2 | |
E6 | A1 | m=T⊕A1⊕A3⊕p1⊕p2⊕p3 | 10 |
A3 | m=T⊕A1⊕A1⊕p1⊕p2⊕p3 | 4 | |
A1×A3 | m=T⊕A1⊕p1⊕p2⊕p3 | 4 | |
A1×A1 | m=T⊕A3⊕p1⊕p2⊕p3 | 2 | |
A1×A1×A3 | m=T⊕p1⊕p2⊕p3 | 1 |