A Kac-Moody algebra associated to the non-compact manifold SL$(2, \mathbb R)$

TL;DR

This paper constructs a Kac-Moody algebra linked to the non-compact manifold SL(2, R) using two methods, revealing its structure through Hilbert bases and the Plancherel theorem, and identifying central extensions and differential operators.

Contribution

It provides an explicit construction of a Kac-Moody algebra associated with SL(2, R) via two equivalent approaches, enhancing understanding of its algebraic and analytical properties.

Findings

Explicit construction of the Kac-Moody algebra for SL(2, R)

Identification of central extensions and Hermitean differential operators

Demonstration of equivalence between two construction methods

Abstract

We construct explicitly a Kac-Moody algebra associated to SL$(2, \mathbb R)$ in two different but equivalent ways: either by identifying a Hilbert basis of $L^2($SL$(2, \mathbb R))$ or by the Plancherel Theorem. Central extensions and Hermitean differential operators are identified.