Pseudo-symmetric pairs for Kac-Moody algebras

TL;DR

This paper introduces pseudo-involutions and pseudo-fixed-point subalgebras in symmetrizable Kac-Moody algebras, generalizing classical involution concepts and extending the theory of symmetric spaces.

Contribution

It defines pseudo-involutions acting on stable Cartan subalgebras and explores their associated structures, extending the framework of symmetric spaces to a broader Kac-Moody context.

Findings

Defined pseudo-involutions of the second kind

Analyzed associated pseudo-fixed-point subalgebras

Extended Satake diagram concepts to Kac-Moody algebras

Abstract

Lie algebra involutions and their fixed-point subalgebras give rise to symmetric spaces and real forms of complex Lie algebras, and are well-studied in the context of symmetrizable Kac-Moody algebras. In this paper we study a generalization. Namely, we introduce the concept of a pseudo-involution, an automorphism which is only required to act involutively on a stable Cartan subalgebra, and the concept of a pseudo-fixed-point subalgebra, a natural substitute for the fixed-point subalgebra. In the symmetrizable Kac-Moody setting, we give a comprehensive discussion of pseudo-involutions of the second kind, the associated pseudo-fixed-point subalgebras, restricted root systems and Weyl groups, in terms of generalizations of Satake diagrams.