Non-abelian GKM Theory

TL;DR

This paper extends GKM theory to actions of arbitrary compact connected Lie groups, using a graph-based approach to encode the non-abelian 1-skeleton and determine the equivariant cohomology algebra.

Contribution

It introduces a non-abelian GKM framework that generalizes the classical abelian case, capturing the structure of non-abelian 1-skeletons and their impact on equivariant cohomology.

Findings

The algebra of equivariant cohomology can be derived from the non-abelian GKM graph.

Non-abelian GKM theory reveals unique features due to the complex structure of the non-abelian 1-skeleton.

The approach generalizes existing GKM methods to broader group actions.

Abstract

We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting group. We show that the algebra structure of the equivariant cohomology can be read off from this graph. In comparison with ordinary abelian GKM theory, there are some special features due to the more complicated structure of the non-abelian 1-skeleton.