An overview of generalised Kac-Moody algebras on compact real manifolds

TL;DR

This paper introduces a generalized framework for Kac-Moody algebras on compact real manifolds, utilizing Fourier analysis and representation theory, and explores their central extensions and applications in physics.

Contribution

It develops a new definition of generalized Kac-Moody algebras on manifolds, constructs Hilbert bases via Fourier expansion, and analyzes their central extensions and root systems.

Findings

Hilbert bases characterized by compact Lie group representations

Existence of central extensions proven using Hermitian operator duality

Applications discussed for physically relevant compact groups

Abstract

A generalised notion of Kac-Moody algebra is defined using smooth maps from a compact real manifold $\mathcal{M}$ to a finite-dimensional Lie group, by means of complete orthonormal bases for a Hermitian inner product on the manifold and a Fourier expansion. The Peter--Weyl theorem for the case of manifolds related to compact Lie groups and coset spaces is discussed, and appropriate Hilbert bases for the space $L^{2}(\mathcal{M})$ of square-integrable functions are constructed. It is shown that such bases are characterised by the representation theory of the compact Lie group, from which a complete set of labelling operator is obtained. The existence of central extensions of generalised Kac-Moody algebras is analysed using a duality property of Hermitian operators on the manifold, and the corresponding root systems are constructed. Several applications of physically relevant compact groups and coset spaces are discussed.