Distinguished orbits and the L-S category of simply connected compact Lie groups

TL;DR

This paper establishes an upper bound for the Lusternik-Schnirelmann category of simply connected compact Lie groups using the relative categories of specific conjugacy classes related to the fundamental alcove and affine Weyl group actions.

Contribution

It introduces a novel approach to bounding the LS-category of Lie groups through the analysis of distinguished conjugacy classes and their relation to the affine Weyl group.

Findings

LS-category of G is bounded above by sum of relative categories of conjugacy classes

Connection between fundamental alcove vertices and conjugacy classes

Provides a new method to estimate topological complexity of Lie groups

Abstract

We show that the Lusternik-Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of G.