Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras

TL;DR

This paper introduces a general framework for -commuting varieties linked to pairs of commuting involutions in semisimple Lie algebras, revealing connections to simple Jordan algebras and analyzing their geometric properties.

Contribution

It develops a comprehensive theory of -commuting varieties, identifies cases with favorable properties, and establishes a link between these varieties and simple Jordan algebras.

Findings

-commuting variety can be isomorphic to the commuting variety of a simple Jordan algebra.

The product map in the Jordan algebra of symmetric matrices is equidimensional.

Equidimensionality fails for all other simple Jordan algebras.

Abstract

We study certain "\sigma-commuting varieties" associated with a pair of commuting involutions of a semisimple Lie algebra $\g$. The usual commuting variety of $\g$ and commuting varieties related to one involution are particular cases of our construction. We develop a general theory of \sigma-commuting varieties and point out some cases, when they have especially good properties. We show that, for a special choice of commuting involutions, the \sigma-commuting variety is isomorphic to the commuting variety of a simple Jordan algebra. As a by-product of our theory, we show that if $J$ is the Jordan algebra of symmetric matrices, then the product map $J \times J\to J$ is equidimensional; while for all other simple Jordan algebras equidimensionality fails.