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Chrysikos, Ioannis; Sakane, Yusuke

doi:10.1016/j.bulsci.2013.11.002

Cited by 11 publications

(30 citation statements)

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“…In this case 2 −1 ν ∈ {1, 3, 10, 41, 172}, and we have ε = ν for 2 −1 ν ∈ {1, 3, 10}. By the recent calculations of I.Chrysikos and Y.Sakane [14] it implies that for G/H = E 8 /T 1 · A 3 · A 4 all complex solutions are isolated, and ε = 81, so that ν − ε = 82 − 81 = 1. The missing solution with multiplicity 1 'escape to infinity'.…”

Section: Introductionmentioning

confidence: 60%

On the compactness of the set of invariant Einstein metrics

Граев

2013

Ann Glob Anal Geom

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Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$. We will assume that the isotropy $H$-module $\mathfrak {g/h}$ has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent. There exists a convex Newton polytope $N=N(G,H)$, which was used for the estimation of the number of isolated complex solutions of the algebraic Einstein equation for invariant metrics on $G/H$ (up to scaling). Using the moment map, we identify the space $\mathcal{M}_1$ of invariant Riemannian metrics of volume 1 on $G/H$ with the interior of this polytope $N$. We associate with a point ${x \in \partial N}$ of the boundary a homogeneous Riemannian space (in general, only local) and we extend the Einstein equation to $\bar{\mathcal{M}_1}= N$. As an application of the Aleksevsky--Kimel'fel'd theorem, we prove that all solutions of the Einstein equation associated with points of the boundary are locally Euclidean. We describe explicitly the set $T\subset \partial N$ of solutions at the boundary together with its natural triangulation. Investigating the compactification $\bar{\mathcal{M}_1} $ of $\mathcal{M}_1$, we get an algebraic proof of the deep result by B\"ohm, Wang and Ziller about the compactness of the set $ \mathcal{E}_1 \subset \mathcal{M}_1$ of Einstein metrics. The original proof by B\"ohm, Wang and Ziller was based on a different approach and did not use the simplicity of the spectrum. In Appendix we consider the non-symmetric K\"ahler homogeneous spaces $G/H$ with the second Betti number $b_2=1$. We write the normalized volumes $2,6,20,82,344$ of the corresponding Newton polytopes and discuss the number of complex solutions of the algebraic Einstein equation and the finiteness problem.Comment: 25 pages, 4 figures. Some proofs, 3 references, and Appendix adde

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Section: Introductionmentioning

confidence: 60%

On the compactness of the set of invariant Einstein metrics

Граев

2013

Ann Glob Anal Geom

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“…Flag manifolds with second Betti number b 2 (M ) = 1 and Lie groups os Type I, II, or III Let G be a compact connected simple Lie group with finite centre and Dynkin diagram Γ(Π), where Π denotes a basis of simple roots. We are interested in K-principal fibrations of G over flag manifolds M = G/K with the aim to build left-invariant metrics on G via a metric on the base G/K and a metric on the fiber K. For a Lie theoretic description of flag manifolds in terms of painted Dynkin diagrams we refer to [AP,AC,C2,CS].…”

Section: Invariant Metrics and The Ricci Tensormentioning

confidence: 99%

“…Lie groups of Type I, II or III. It is well-known (see [C2,C1,CS]) that by painting black on Γ(Π) the vertex of a simple root, say α io for some 1 ≤ i o ≤ ℓ, we define a flag manifold M = G/K whose isotropy representation decomposes into q := ht(α io ) ∈ Z + mutually inequivalent, irreducible Ad(K)-submodules, i.e. p = T o M = p 1 ⊕ · · · ⊕ p q .…”

Section: Invariant Metrics and The Ricci Tensormentioning

confidence: 99%

“…These inclusions occur since the base space of the K-principal bundle G → M = G/K is a flag manifold with p = p 1 ⊕ p 2 ⊕ p 3 (= m 2 ⊕ m 3 ⊕ m 4 ) and b 2 (M ) = 1 (see [AnC, p. 1593] and [CS,p. 674] for the general case).…”

Section: 2mentioning

confidence: 99%

“…Finally based on Propositions 4.6, 5.5, 6.4, we deduce which of these solutions induce new non-naturally reductive left-invariant Einstein metrics. We summarize our results (up to isometry) in the following theorem (for useful details on the examination of the isometry problem we refer to [CS,AC,C2]).…”

Section: Introductionmentioning

confidence: 99%

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Non-naturally reductive Einstein metrics on exceptional Lie groups

Chrysikos

Sakane

2017

Journal of Geometry and Physics

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Given an exceptional compact simple Lie group G we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of G over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-invariant metrics. Then we apply a technique based on Gröbner bases and classify the real solutions of the associated algebraic systems. For the Lie group G 2 we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, for the Lie groups E 7 and E 8 , we conclude that there exist non-isometric non-naturally reductive Einstein metrics, which are Ad(K)-invariant by different Lie subgroups K.

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Spin Structures on Compact Homogeneous Pseudo-Riemannian Manifolds

Alekseevsky

Chrysikos

2018

Transformation Groups

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We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds F = G/H, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M = G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F . We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.2000 Mathematics Subject Classification. 53C10, 53C30, 53C50, 53D05. Keywords: spin structure, metaplectic structure, homogeneous pseudo-Riemannian manifold, flag manifold, Koszul form, C-spaceDedicated to the memory of M. Graev 1 2 DMITRI V. ALEKSEEVSKY AND IOANNIS CHRYSIKOS Our results can be read as follows. After recalling some basic material in Section 1, in Section 2 we study invariant spin structures on pseudo-Riemannian homogeneous spaces, using homogeneous fibrations. Recall that given a smooth fibre bundle π : E → B with connected fibre F , the tangent bundle T F of F is stably equivalent to i * (T E), where i : F ֒→ E is the inclusion map (cf. [Sin]). Evaluating this result at the level of characteristic classes, one can treat the existence of a spin structure on the total space E in terms of Stiefel-Whitney classes of B and F , in the spirit of the theory developed by Borel and Hirzebruch [BoHi]. We apply these considerations for fibrations induced by a tower of closed connected Lie subgroups L ⊂ H ⊂ G (Proposition 2.6) and we describe sufficient and necessary conditions for the existence of a spin structure on the associated total space (Corollary 2.7, see also [GGO]). Next we apply these results in several particular cases. For example, in Section 3 we classify spin and metaplectic structures on compact homogeneous Kähler manifolds of a compact connected semisimple Lie group G, i.e. (generalized) flag manifolds.Generalized flag manifolds are homogeneous spaces of the form G/H, where H is the centralizer of torus in G. Here, we explain how the existence and classification of invariant spin or metaplectic structures can be treated in term of representation theory (painted Dynkin diagrams) and provide a criterion in terms of the so-called Koszul numbers (Proposition 3.12). These are the integer coordinates of the invariant Chern form (which represent the first Chern class of of an invariant complex structure J of F = G/H), with respect to the fundamental weights. By applying an algorithm given in [AℓP] (slightly revised), we compute the Koszul numbers for any flag manifold corresponding to a classical Lie group and provide necessary and sufficient conditions for the existence of a spin or metaplecti...

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The classification of homogeneous Einstein metrics on flag manifolds withb2(M)=1

Cited by 11 publications

References 20 publications

On the compactness of the set of invariant Einstein metrics

On the compactness of the set of invariant Einstein metrics

Non-naturally reductive Einstein metrics on exceptional Lie groups

Spin Structures on Compact Homogeneous Pseudo-Riemannian Manifolds

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