An infinite-rank Lie algebra associated to SL$(2,\mathbb R)$ and SL$(2,\mathbb R)/U(1)$

TL;DR

This paper constructs a new class of infinite-rank Lie algebras based on smooth maps from non-compact manifolds related to SL(2,R), with potential applications in supergravity and a detailed analysis of their algebraic structure.

Contribution

It introduces a novel generalization of Kac-Moody algebras using non-compact manifolds and explores their properties, including central extensions and root structures.

Findings

Defined Lie brackets for the generalized algebras

Identified the root structure and commuting operators

Discussed applications in supergravity scenarios

Abstract

We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds ${\cal M}=$SL$(2,\mathbb R)$ and ${\cal M}=$ SL$(2,\mathbb R)/U(1)$ to a finite-dimensional simple Lie group $G$. This construction is achieved through two equivalent ways: by means of the Plancherel Theorem and by identifying a Hilbert basis within $L^2(\mathcal{M})$. We analyse the existence of central extensions and identify those in duality with Hermitean operators on $\cal M$. By inspecting the Clebsch-Gordan coefficients of $\mathfrak{sl}(2,\mathbb{R})$, we derive the Lie brackets characterising the corresponding generalised Kac-Moody algebras. The root structure of these algebras is identified, and it is shown that an infinite number of simultaneously commuting operators can be defined. Furthermore, we briefly touch upon applications of these algebras within the realm of supergravity, particularly in scenarios where the scalar fields coordinatize the non-compact manifold $\text{SL}(2,\mathbb{R})/U(1)$.