Some automorphisms of Generalized Kac-Moody algebras
J"urgen Fuchs, Urmie Ray, Christoph Schweigert

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic structures and combinatorial models · Nonlinear Waves and Solitons · Advanced Topics in Algebra
