Invariant Einstein Metrics on SU(4)/T

Transcription

1 43 5ff fl ± ν Vol.43 No Ω9» ADVANCES IN MATHEMATICS(CHINA) Sep doi: /sxjz b Invariant Einstein Metrics on SU(4)/T WANG Yu LI Tianzeng (School of Science Sichuan University of Science and Engineering Zigong Sichuan P. R. China) Abstract: We compute non-zero structure constants of the classical full flag manifold M = SU(4)/T with six isotropy summands then construct the Einstein equation. It is well known that there are 29 SU(4)-invariant Einstein metrics (up to a scale) on M = SU(4)/T. With the help of computer we get all the 29 positive solutions (up to a scale) of the system of Einstein equation for the full flag manifold SU(4)/T whereoneiskähler-einstein metric (up to isometry) and three are non-kähler-einstein metrics (up to isometry). Keywords: generalized flag manifold; Einstein metric; Ricci tensor; isotropy representation MR(2010) Subject Classification: 53C25 / CLC number: O186 Document code: A Article ID: (2014) Introduction An important class of homogeneous manifolds is formed by the orbits of the adjoint action of a semisimple compact Lie Group called generalized flag manifolds. Such manifolds can be described by a quotient M = G/C(T ) where C(T ) is the centralizer of a torus T of the Lie group G. IfC(T )=T then M = G/T is called a full flag manifold. Non-Kähler-Einstein metrics on full flag manifolds corresponding to classical Lie groups have been studied by several authors [4 8 11]. But when the isotropy representation of the full flag manifolds increases it is very difficult to find all the non-kähler-einstein metrics (up to a scale) since there is no good method to get all the positive solutions of the system of Einstein equations. According to [9] we know that there are 29 positive solutions for the system of Einstein equations of SU(4)/T but it is very difficult for us to get all the positive solutions. Arvanitoyeorgos Chrysiko and Sakane [6] used the method of computing the Gröbner basis to get the positive solutions of the system of Einstein equations for some generalized flag manifolds. But when the isotropy representation of the flag manifolds increases it is very difficult to compute the Gröbner basis of the system of Einstein equations. Thus invariant Einstein metrics on the generalized flag manifolds with more than five isotropy summands remains an open problem. In this paper we use the command NSolve in Mathematica to find the positive solutions of the system of Einstein equations for SU(4)/T. Using this method we get all the 29 positive solutions of the system of Einstein equations for the full flag manifold SU(4)/T (up to a scale). Received date: Foundation item: Partially supported by Scientific Research Fund of Sichuan University of Science and Engineering (No. 2012PY17) Scientific Research Fund of Sichuan Provincial Education Department (No. 14ZB0173) and Artificial Intelligence Key Laboratory of Sichuan Province (No. 2014RYJ05). wangyu 813@163.com; litianzeng27@163.com

2 782 fi Π μ 43 The full flag manifold SU(4)/T wheret = U(1) U(1) U(1) is a maximal torus in SU(4) is obtained by painting black three simple roots in the Dynkin diagram of SU(4). According to [13] a full flag manifold G/T is a normal homogeneous Einstein manifold if and only if all of the roots of G have the same length and in this case the normal metric of G/T is never Kähler. This paper is organized as follows. In Section 1 we recall the Lie-theoretic description of a generalized flag manifold G/K of a compact and connected semisimple Lie group G. In Section 2 we consider the classical full flag manifold SU(4)/T and give its Lie-theoretic description and give 29 SU(4)-invariant Einstein metrics for the full flag manifold SU(4)/T (up to a scale) then we prove that SU(4)/T admits exactly four SU(4)-invariant Einstein metrics (up to isometry). 1 Generalized Flag Manifold In this section we recall the Lie-theoretic description of M = G/K. Let k and g be the Lie algebras of K and G respectively and ( ) be the Cartan Killing form on the Lie algebra g. Let g C and k C be the complexifications of g and k respectively. The complexification h C is a Cartan subalgebra of g C whereh is the Cartan subalgebra of g. We denote by (h C ) the dual space of h C and by R (h C ) the root system of g C relative to h C. We consider the root space decomposition g C = h C α R gc α. Choose a simple root system Π = {α 1 α 2 α l } (dim h = l) ofr. We fix a lexicographic ordering on (h C ) and let R + be the set of positive roots with respect to Π. Choose a subset Π K of Π and set Π M =Π\Π K = {α i1 α i2 α ir } (1 i 1 i 2 i r l). Let R K = R Π K R + K = R+ Π K R + M = R+ \R + K (1.1) where Π K denotes the set of roots generated by Π K.ThesetR M is such that R = R K R M and is called the set of complementary roots of M = G/K. We choose a Weyl basis E α g C α (α R) ofg C with (E α E α )=1 [E α E α ]=H α and { 0 if α + β R; [E α E β ]= N αβ E α+β if α + β R (1.2) where the constants N αβ are such that N αβ = N α β and N βα = N αβ. Then we obtain that g = h (RA α + RB α ) (1.3) α R + where A α = E α E α B α =i(e α + E α ) α R +. The subalgebra p = h C g C α g C α (1.4) α R K α R + M is a parabolic subalgebra of g C. The intersection k = p g g is the Lie subalgebra corresponding to K and is given by k = h α R + K (RA α + RB α ). Let m = (RA α + RB α ). (1.5) α R + M

3 5ff ß Ψffl : Invariant Einstein Metrics on SU(4)/T 783 According to (1.3) and (1.4) we easily obtain the direct decomposition p = k C m where k C = h C α R K g C α and m C = α R + M g C α. For convenience we fix a system of simple roots Π = {α 1 α 2 α r φ 1 φ 2 φ k } of R so that Π K = {φ 1 φ 2 φ k } is a basis of the root system R K and Π M = Π \ Π K = {α 1 α 2 α r } (r + k = l). We consider the decomposition R = R K R M and define the set t = z(k C ) ih = {X h : φ(x) =0 φ R K } where z presents the center of k C. Consider the linear restriction map κ : h t defined by κ(α) =α t andsetr t = κ(r) =κ(r M ). Note that κ(r K )=0andκ(0) = 0. Definition 1.1 The elements of R t are called t-roots. For an invariant ordering R + M = R+ \ R + K in R M wesetr t + = κ(r + M )andr t = R t + = { ξ : ξ R t + }. It is obvious that Rt = κ(r M ) thus the splitting R t = Rt R t + defines an ordering in R t.thet-roots ξ R t + (resp. ξ Rt ) are called positive (resp. negative). [2 Proposition 4.1] Proposition 1.1 There is one-to-one correspondence between t-roots and complex irreducible ad(k C )-submodules m ξ of m C. This correspondence is given by R t ξ m ξ = CE α. α R M :κ(α)=ξ Thus m C = ξ R t m ξ. Moreover these submodules are inequivalent as ad(k C )-modules. Since the complex conjugation τ : g C g C X +iy X iy (X Y g) ofg C with respect to the compact real form g interchanges the root spaces i.e. τ(e α )=E α and τ(e α )=E α a decomposition of the real ad(k)-module m =(m C ) τ into real irreducible ad(k)-submodule is given by m = (m ξ m ξ ) τ (1.6) ξ R + =κ(r + M ) where n τ denotes the set of fixed points of the complex conjugation τ in a vector subspace n g C. If for simplicity we set R t + = {ξ 1 ξ 2 ξ s } then according to (1.6) each real irreducible ad(k)- submodule m i =(m ξi m ξi ) τ (1 i s) corresponding to the positive t-root ξ i isgiven by m i = RA α + RB α. (1.7) α R + M Definition 1.2 A t-root is called simple if it is not a sum of two positive t-roots. Proposition 1.2 [5 Proposition 4] Let Π M =Π\ Π K = {α 1 α 2 α r }. Then the set {ᾱ i = α i t : α i Π M } is a t-base of t. A G-invariant Riemannian metric g on M is identified with an Ad(K)-invariant inner product on m which can be written as X Y = (ΛX Y )(X Y m) where Λ : m m is an Ad(K)-invariant positive definite symmetric endomorphism on m. Due to the decomposition (1.6) we can express Λ as Λ = ξ R + x ξ Id (mξ m t ξ ) τ where each element in {x ξ : ξ R t + } is an eigenvalue of Λ. Due to the decomposition (1.7) Λ is given by Λ= s x ξi Id mi = x ξi Id mi where x i x ξi for any ξ i R t + = {ξ 1 ξ 2 ξ s }. (1.8) ξ i R + t i=1

4 784 fi Π μ 43 Due to (1.7) it is obvious that the vectors {A α B α : α R + M } are eigenvectors of Λ corresponding to the eigenvalue x i x ξi. We also denote this eigenvalue by x α R + where α R + M is such that κ(α) =ξ i for any 1 i s. We extend Λ to m C without any change in notation. Hence the inner product g = admits a natural extension to an ad(k C )-invariant bilinear symmetric form on m C. Then the root vectors {E α : α R M } are eigenvectors of Λ:m C m C corresponding to the eigenvalues x α = x α > 0. If we denote by {ω α } the basis of the dual space(m C ) which is dual to the basis {E β β R M } i.e. ω α (E β )=δβ αthenwe obtain that [1 3] Proposition 1.3 Any real ad(k C )-invariant inner product g = on m C has the form g = = x α ω α ω β = ω α ω β (1.9) α R + M x ξ ξ R + α κ (ξ) t where ω α ω β = 1 2 (ωα ω β + ω β ω α ) and the positive real numbers x α are such that x α = x β if α t = β t for any α β R + M. The space of G-invariant Riemannian metrics g = = (Λ )onm is given by {x 1 ( ( )) m1 +{x 2 ( ( )) m2 + + x s ( ( )) ms : x 1 > 0x 2 > 0 x s > 0} (1.10) where x 1 x ξ1 > 0 x 2 x ξ2 > 0 x s x ξs > 0. Then the Ricci tensor Ric g of G/K asag-invariant symmetric covariant 2-tensor on G/K is identified with an Ad(K)-invariant symmetric bilinear form on m given by Ric g = γ 1 x 1 ( ( )) m1 +γ 2 x 2 ( ( )) m2 + + γ s x s ( ( )) ms. (1.11) Here γ 1 γ 2 γ s are the components of the Ricci tensor on each m i. Proposition 1.4 [7] Let g = be a G-invariant metric given by (1.9) and J be a G-invariant complex structure induced by an invariant ordering R + M.Thengis a Kähler metric with respect to the complex structure J if and only if the positive real numbers x ξ satisfy x ξ+ζ = x ξ + x ζ for any ξζξ + ζ R t + = κ(r + M ). Equivalently g is Kähler if and only if x α+β = x α + x β where α β α + β R + M are such that κ(α) =ξ and κ(β) =ζ. Let {e α } be an orthogonal basis with respect to ( ) adapted to the decomposition of m: e α m i and e β m j with i<jthen α<β. Following [13] we set A γ αβ := ([e αe β ]e γ ) and thus [e α e β ] m = γ Aγ αβ e γ. Consider c k ij := (A γ αβ )2 (1.12) where the sum is taken over all indices α β γ with e α m i e β m j e γ m k and i j k {1 2 s}. Hence c k ij is nonnegative symmetric in all the three entries and is independent of the orthogonal basis chosen for m i m j and m k. But it depends on the choice of the decomposition of m. We now introduce the notion of symmetric t-triples of t-roots. Definition 1.3 A symmetric t-triple in t is a triple Ω = (ξ i ξ j ξ k )of t-roots ξ i ξ j ξ k R t such that ξ i + ξ j + ξ k =0.

5 5ff ß Ψffl : Invariant Einstein Metrics on SU(4)/T 785 [9 unpublished] Lemma 1.1 Let (ξ i ξ j ξ k ) be symmetric t-triples. Then there exist roots α β γ R M with κ(α) =ξ i κ(β) =ξ j κ(γ) =ξ k such that α + β + γ =0. The calculation of the coefficients c k ij can be laborious. However the next result shows exactly which of them are non-zero. [7 Corollary 1.9] Lemma 1.2 Let G/K be a generalized flag manifold of a compact simple Lie group G and R t be the associated t-root system. Assume that m = m 1 m 2 m s is a ( )-orthogonal decomposition of m into pairwise inequivalent irreducible Ad(k)-modules and let ξ i ξ j ξ k R t be the t-root associated to the components m i m j and m k respectively. Then c k ij 0 if and only if (ξ iξ j ξ k ) is a symmetric t-triple i.e. ξ i + ξ j + ξ k =0. 2 Invariant Einstein Metrics on SU(4)/T Lemma 2.1 [6] For a full flag manifold G/T the non-zero structure constant c k ij is given by c k ij =(Aα+β αβ )2 =2Nαβ 2 (2.1) where α β R + with κ(α) =ξ i κ(β) =ξ j κ(α + β) =ξ k. Lemma 2.2 [10] Let M = G/K be a reductive homogeneous space of a compact semisimple Lie group G and let m = m 1 m 2 m s be a decomposition of m into mutually inequivalent irreducible Ad(K)-submodules. Then the components γ 1 γ 2 γ s of the Ricci tensor of a G-invariant metric (1.10) on M are given by γ k = 1 2x k + 1 4d k ij x k c k ij 1 x i x j 2d k ij x j c j ki k =1 2 s (2.2) x k x i where d i = dim(m i ). Next we consider full flag manifold of SU(4)/T defined by painted Dynkin graph α 1 α 2 Here Π M = {α 1 α 2 α 3 }. Let ᾱ 1 = κ(α 1 ) ᾱ 2 = κ(α 2 )andᾱ 3 = κ(α 3 ) we have R t + = κ(r + M )= {ᾱ 1 ᾱ 2 ᾱ 3 ᾱ 1 +ᾱ 2 ᾱ 2 +ᾱ 3 ᾱ 1 +ᾱ 2 +ᾱ 3 } and thus we conclude the isotropy representation m = 6 i=1 m i. By Lemma 1.2 we obtain that the non-zero structure constants are c 4 12 c6 15 c5 23 c6 34. Lemma 2.3 The non-zero structure constants c 4 12c 5 23c 6 15c 6 34 of generalized flag manifold SU(4)/T with respect to the decomposition m = 6 i=1 m i are given by c 4 12 = c 6 15 = c 5 23 = c 6 34 = 1 4. Proof We know that Nαβ 2 = q(p+1) 2 (α α) (α β) = q p 2 (α α) where p q are the largest nonnegative integers such that β + kα R with p k q and( ) is the Cartan-Killing form on the Lie algebra g (see [12 p. 286]). By Lemma 2.1 we can calculate the non-zero structure constants of M as follows: α 3. c 4 12 =2Nα 2 1α 2 =(α 1 α 1 ) c 6 15 =2N α 2 1α 2+α 3 =(α 1 α 1 ) c 5 23 =2Nα 2 2α 3 =(α 1 α 1 ) c 6 34 =2Nα 2 3α 1+α 2 =(α 1 α 1 ).

6 786 fi Π μ 43 As (α 1 α 1 )= 1 4 we obtain that c4 12 = c6 15 = c5 23 = c6 34 = 1 4. Lemma 2.4 The components γ i (i =1 2 6) of Ricci tensor associated to the SU(4)- invariant Riemannian metric g are the followings: γ 1 = 1 + x2 1 x2 2 x2 4 + x2 1 x2 5 x2 6 2x 1 16x 1 x 2 x 4 16x 1 x 5 x 6 γ 2 = 1 + x2 2 x 2 1 x x2 2 x 2 3 x 2 5 2x 2 16x 1 x 2 x 4 16x 2 x 3 x 5 γ 3 = 1 + x2 3 x2 4 x2 6 + x2 3 x2 2 x2 5 2x 3 16x 3 x 4 x 6 16x 3 x 2 x 5 γ 4 = 1 + x2 4 x2 1 x2 2 + x2 4 x2 3 x2 6 2x 4 16x 1 x 2 x 4 16x 3 x 4 x 6 γ 5 = 1 + x2 5 x 2 1 x x2 5 x 2 2 x 2 3 2x 5 16x 1 x 5 x 6 16x 2 x 3 x 5 γ 6 = 1 + x2 6 x2 1 x2 5 + x2 6 x2 3 x2 4. 2x 6 16x 1 x 5 x 6 16x 3 x 4 x 6 Proof By substituting the dimension d i = dim(m i ) = 2 and the non-zero structure constants c k ij into (2.2) we can get the results. From (1.10) and (1.11) we get that a G-invariant Riemannian metric g on M = SU(4)/T is Einstein if and only if there is a positive constant e such that γ 1 = γ 2 = γ 3 = γ 4 = γ 5 = γ 6 = e or equivalently γ 1 γ 2 =0 γ 2 γ 3 =0 γ 3 γ 4 =0 γ 4 γ 5 =0 γ 5 γ 6 =0. (2.3) By Lemma 2.4 and the system (2.3) we obtain the following polynomial system (we apply the normalization x 1 =1) x 2 3 x 4x 6 x 2 2 x 4x 6 + x 2 5 x 4x 6 + x 2 x 3 x 4 +2x 3 x 5 x 6 8x 3 x 4 x 5 x 6 x 2 x 3 x 4 x 2 5 x 2x 3 x 4 x 2 6 2x 3 x 2 2x 5 x 6 +8x 2 x 3 x 4 x 5 x 6 =0; x 2 2 x 3x 5 x 6 +2x 2 2 x 4x 6 x 2 3 x 2x 5 + x 2 4 x 2x 5 8x 2 x 4 x 5 x 6 + x 2 6 x 2x 5 2x 2 3 x 4x 6 x 2 4 x 3x 5 x 6 +8x 3 x 4 x 5 x 6 x 5 x 3 x 6 =0; x 2 2 x 3x 5 x 6 x 2 2 x 4x 6 +2x 2 x 2 3 x 5 8x 2 x 3 x 5 x 6 2x 2 x 2 4 x 5 +8x 2 x 4 x 5 x 6 + x 2 3 x 6x 4 x 2 4 x 3x 5 x 6 + x 3 x 5 x 6 x 4 x 2 5x 6 =0; x 2 2 x 3x 5 x 6 + x 2 2 x 4x 6 x 2 x 5 x 2 3 x 2x 3 x 4 x x 2x 3 x 4 x 2 6 8x 2x 3 x 4 x 6 + x 2 x 3 x 4 +8x 2 x 3 x 5 x 6 + x 2 x 2 4x 5 x 2 x 5 x x 2 3x 4 x 6 + x 3 x 2 4x 5 x 6 x 5 x 3 x 6 x 4 x 2 5x 6 =0; x 2 2 x 4x 6 + x 2 x 2 3 x 5 +2x 2 x 3 x 4 x 2 5 8x 2x 3 x 4 x 5 2x 2 x 3 x 4 x x 2x 3 x 4 x 6 + x 2 x 2 4 x 6 x 2 x 5 x 2 6 x 2 3 x 4x 6 + x 4 x 2 5 x 6 =0. Any positive real solution x 2 x 3 x 6 > 0 of the system above determines a SU(4)- invariant Einstein metric (1x 2 x 3 x 4 x 5 x 6 ) R 6 + on M = G/T.WeusethecommandNSolve in Mathematica and obtain all the 29 positive solutions. Theorem 2.1 The flag manifold M = SU(4)/T admits precisely 29 SU(4)-invariant Ein-

7 5ff ß Ψffl : Invariant Einstein Metrics on SU(4)/T 787 stein metrics (up to a scale) which approximately are given as follows: (1) ( ) (2) ( ) (3) ( ) (4) ( ) (5) ( ) (6) ( ) (7) ( ) (8) ( ) (9) ( ) (10) ( ) (11) ( ) (12) ( ) (13) ( ) (14) ( ) (15) ( ) (16) ( ) (17) ( ) (18) ( ) (19) ( ) (20) ( ) (21) ( ) (22) ( ) (23) ( ) (24) ( ) (25) ( ) (26) ( ) (27) ( ) (28) ( ) (29) ( ) where (1) is the unique Kähler-Einstein metric. Note that the action of the Weyl group of SU(4) on the root system of SU(4) induces an action on the components of the SU(4)-invariant metric (1.10). In particular if (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 1 a 2 a 3 a 4 a 5 a 6 ) is a solution of the system to the equation above then (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 1 a 4 a 3 a 2 a 6 a 5 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 1 a 5 a 3 a 6 a 2 a 4 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 1 a 6 a 3 a 5 a 4 a 2 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 2 a 1 a 6 a 4 a 5 a 3 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 2 a 4 a 6 a 1 a 3 a 5 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 2 a 3 a 6 a 5 a 4 a 1 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 2 a 5 a 6 a 3 a 1 a 4 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 6 a 3 a 2 a 4 a 5 a 1 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 6 a 1 a 2 a 5 a 4 a 3 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 6 a 5 a 2 a 1 a 3 a 4 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 6 a 4 a 2 a 3 a 1 a 5 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 5 a 2 a 4 a 3 a 1 a 6 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 5 a 3 a 4 a 2 a 6 a 1 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 5 a 1 a 4 a 6 a 2 a 3 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 5 a 6 a 4 a 1 a 3 a 2 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 4 a 6 a 5 a 3 a 1 a 2 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 4 a 3 a 5 a 6 a 2 a 1 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 4 a 2 a 5 a 1 a 3 a 6 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 4 a 1 a 5 a 2 a 6 a 3 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 3 a 6 a 1 a 4 a 5 a 2 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 3 a 5 a 1 a 2 a 6 a 4 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 3 a 4 a 1 a 6 a 2 a 5 ) (x 1 x 2 x 3 x 4 x 5 x 6 )=(a 3 a 2 a 1 a 5 a 4 a 6 ) are also the solutions to the system above. These metrics are all isometric to each other.

8 788 fi Π μ 43 Theorem 2.2 The flag manifold M = SU(4)/T admits exactly four SU(4)-invariant Einstein metrics (up to a isometry). There is a unique Kähler Einstein metric (up to a scale) given by g =( ) and the other three are non-kähler (up to a scale) given by: (a) ( ) (b) ( ) (c) ( ). References [1] Alekseevsky D.V. Homogeneous Einstein metrics Diff. Geom. Appl. In: Proc. Conf. Brno [2] Alekseevsky D.V. Flag manifolds In: Sbornik Radova 11th Jugoslav Geom. Seminar. Beograd (14): [3] Alekseevsky D.V. and Perelomov A.M. Invariant Kähler-Einstein metrics on compact homogeneous spaces Funct. Anal. Appl (3): [4] Arvanitoyeorgos A. New invariant Einstein metrics on generalizd flag manifolds Trans. Amer. Math. Soc (2): [5] Arvanitoyeorgos A. and Chrysikos I. Invariant Einstein metrics on generalized flag manifolds with four isotropy summands Ann. Glob. Anal. Geom : [6] Arvanitoyeorgos A. Chrysikos I. and Sakane Y. Homogeneous Einstein metrics on G 2 /T Proc. Amer. Math. Soc (7): [7] Chrysikos I. Flag manifolds symmetric t-triples and Einstein metrics Diff. Geom. Appl (6): [8] Dos Santos E.C.F. and Negreiros C.J.C. Einstein metrics on flag manifolds Revista Della Unión Mathemática Aregetina (2): [9] Graev M.M. On the number of invariant Einstein metrics on a compact homogeneous space Newton polytopes and contraction of Lie algebras Intern.J.Geom.Math.Mod.Phys (5/6): [10] Park J.S. and Sakane Y. Invariant Einstein metrics on certain homogeneous space Tokyo J. Math (1): [11] Sakane Y. Homogeneous Einstein metrics on flag manifold Lobachevskii J. Math : [12] Varadarajan V.S. Lie Group Lie Algebra and Their Representations New York: Springer-Verlag [13] Wang Y. and Ziller W. On normal homogeneous Einstein manifolds Ann. Scient. Éc. Norm. Sup (4): [14] Wang Y. and Ziller W. Existence and non-excistence of homogeneous Einstein metrics Invent. Math : SU(4)/T "fiffρ&#$ffi! * ()+ (ffi ΦΛ Φ οξ ffi ) '% /_E[NRa1XB^ 6 9QTPb M = SU(4)/T W=OF@4Y VC?6 N-eZ]<5. m nk QTPb M = SU(4)/T h34y.9uj`g 29 > SU(4) eZ];M. LfE[D5c 87NQTPb SU(4)/T 9-eZ]<5o9 29 >jg (34Y.9UJ`) Slh:H9UJ`g 1 >IK-eZ];M 3>=IK-eZ];M. χψfl AdTPb; -ez];m; Ricci im; Ra1X