Some automorphisms of Generalized Kac-Moody algebras

TL;DR

This paper explores automorphisms of Generalized Kac-Moody algebras induced by symmetries of their Cartan matrices, introducing twining characters that relate to orbit Lie algebras and their characters.

Contribution

It establishes a connection between automorphisms of GKM algebras, twining characters, and orbit Lie algebras, providing a new perspective on their representation theory.

Findings

Twining characters satisfy a specific character formula.

The subgroup involved is the Weyl group of an orbit Lie algebra.

Twining characters match the characters of the orbit Lie algebra.

Abstract

To any symmetry of the Cartan matrix of a Generalized Kac-Moody (GKM) algebra we associate a family of automorphisms of the algebra which act in a natural way on the modules of the GKM algebra. We introduce the twining character of a module which is obtained by inserting the action of such an automorphism into the trace that appears in the ordinary character. Twining characters of integrable highest weight modules and Verma modules satisfy a character formula which involves a certain subgroup of the Weyl group. This subgroup is shown to be the Weyl group of another GKM algebra, called the orbit Lie algebra, and hence the twining characters coincide with ordinary characters of the orbit Lie algebra.