Flag manifolds, symmetric $\fr{t}$-triples and Einstein metrics

Abstract

Let $G$ be a compact connected simple Lie group and let $M=G^{\bb{C}}/P=G/K$ be a generalized flag manifold. In this article we focus on an important invariant of $G/K$, the so called $\fr{t}$-root system $R_{\fr{t}}$, and we introduce the notion of symmetric $\fr{t}$-triples, that is triples of $\fr{t}$-roots $ξ, ζ, η\in R_{\fr{t}}$ such that $\xi +\eta +\zeta =0$. We describe their properties and we present an interesting application on the structure constants of $G/K$, quantities which are straightforward related to the construction of the homogeneous Einstein metric on $G/K$. Next we classify symmetric $\fr{t}$-triples for generalized flag manifolds $G/K$ with second Betti number $b_2(G/K)=1$, and we treat also the case of full flag manifolds $G/T$, where $T$ is a maximal torus of $G$. In the last section we construct the homogeneous Einstein equation on flag manifolds $G/K$ with five isotropy summands, determined by the simple Lie group $G=\SO(7)$. By solving the corresponding algebraic system we classify all $\SO(7)$-invariant (non-isometric) Einstein metrics, and these are the very first results towards the classification of homogeneous Einstein metrics on flag manifolds with five isotropy summands.

18 pages (the text has been reduced to 18 pages, some misprints has been corrected, unchanged results)

References in corpus (4)

• Invariant Einstein metrics on flag manifolds with four isotropy summands
• The Ricci flow approach to homogeneous Einstein metrics on flag manifolds
• The Ricci flow of left invariant metrics on full flag manifold SU(3)/T from a dynamical systems point of view
• Complete description of invariant Einstein metrics on the generalized flag manifold $SO(2n)/U(p)\times U(n-p)$

Cited by in corpus (4)

• Non-naturally reductive Einstein metrics on exceptional Lie groups
• Homogeneous Einstein metrics on generalized flag manifolds with five isotropy summands
• The classification of homogeneous Einstein metrics on flag manifolds with $b_2(M)=1$
• Homogeneous Einstein metrics on non-Kähler C-spaces