On Formality of Some Homogeneous Spaces

Aleksy Tralle

doi:10.3390/sym11081011

Abstract

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

Highlights

Formality is an important homotopic property of topological spaces
An interesting issue is the formality of homogeneous spaces of compact Lie groups
It should be noted that there is a general method of studying the formality property of homogeneous spaces in terms of the Lie group-theoretic data [3,7]

Summary

Introduction

Formality is an important homotopic property of topological spaces. It is often related to the existence of particular geometric structures on manifolds. It should be noted that there is a general method of studying the formality property of homogeneous spaces in terms of the Lie group-theoretic data [3,7]. In this note we show that if one restricts himself to this class of Riemannian manifolds, the proof can be obtained entirely in terms of the data of the 3-Sasakian homogeneous space G/H (at least for classical Lie groups G). [9] contains much stronger and more general result, the direct proof still may be of independent interest This is motivated by the fact that homogeneous 3-Sasakian manifolds G/H admit a description in terms of the root systems of the complexified Lie algebra gc , and in some cases, the formality property can be expressed via the same data [7] (see [5,6]). The method of proof uses the generators of the ring of invariants of the Weyl group, which becomes computationally difficult (compare, for example the expressions of such polynomials for the exceptional Lie groups [10])

Presentation and Notation

Formality

Quaternionic-Kaehler and 3-Sasakian Manifolds

Cartan Algebras

Application

Proof of Theorem 2

Conclusions