Publication details
Flag manifolds, symmetric t-triples and Einstein metrics
| Authors | |
|---|---|
| Year of publication | 2012 |
| Type | Article in Periodical |
| Magazine / Source | Differential Geometry and its Applications |
| Citation | |
| Web | https://www.sciencedirect.com/science/article/pii/S0926224512000812 |
| Doi | http://dx.doi.org/10.1016/j.difgeo.2012.09.001 |
| Keywords | Generalized flag manifolds; t-Roots; Symmetric t-triples; Root systems; Structure constants; Homogeneous Einstein metrics |
| Description | Let G be a compact connected simple Lie group and let M = G(C)/P = G/K be a generalized flag manifold. In this article we focus on an important invariant of G/K, the so-called t-root system R-t, and we introduce the notion of symmetric t-triples, that is triples of t-roots xi, zeta, eta is an element of R-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. We classify symmetric t-triples for generalized flag manifolds G/K with second Betti number b(2)(G/K) = 1, and next we treat the case of full flag manifolds G/T, with b(2)(G/T)=l= rk G, 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. |