$\mathfrak{k}$-structure of basic representation of affine algebras

TL;DR

This paper explores the structure of basic representations of affine Kac-Moody algebras, revealing a detailed composition series and connections to finite-dimensional representations of maximal compact subalgebras, with applications to supergravity.

Contribution

It introduces a new relation between basic representations and finite-dimensional subrepresentations of maximal compact subalgebras, providing an infinite series and applications.

Findings

Constructed infinitely many $rak{k}$-subrepresentations of the basic representation.

Proved these are all finite-dimensional $rak{k}$-subrepresentations with specific quotient properties.

Presented applications to supergravity theories.

Abstract

This article presents a new relation between the basic representation of split real simply-laced affine Kac-Moody algebras and finite dimensional representations of its maximal compact subalgebra $\mathfrak{k}$. We provide infinitely many $\mathfrak{k}$-subrepresentations of the basic representation and we prove that these are all the finite dimensional $\mathfrak{k}$-subrepresentations of the basic representation such that the quotient of the basic representation by the subrepresentation is a finite dimensional representation of a certain parabolic algebra and of the maximal compact subalgebra. By this result we provide an infinite composition series with a cosocle filtration of the basic representation. Finally, we present examples of the results and applications to supergravity.