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

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.

52 pages, Some comments on generalised Kac-Moody algebras associated to non-compact Lie groups have been added

Cited by in corpus (4)

• Infinite and finite consistent truncations on deformed generalised parallelisations
• Vertex operator for generalised Kac--Moody algebras associated to the two-sphere and the two-torus
• Fermion realisations of generalised Kac--Moody and Virasoro algebras associated to the two-sphere and the two-torus
• Kac-Moody Algebras on Soft Group Manifolds