Representations of involutory subalgebras of affine Kac-Moody algebras

TL;DR

This paper investigates involutory subalgebras of affine Kac-Moody algebras fixed by the Chevalley involution, focusing on their structure and representations, especially in relation to supergravity spinor representations.

Contribution

It introduces a new formulation of these subalgebras as $ $-graded Lie algebras, enabling the construction of a broad class of representations.

Findings

Finite-dimensional unfaithful representations exist.

A new $ $-graded formulation facilitates representation construction.

Connections to supergravity spinor representations are established.

Abstract

We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of $\mathbb{N}$-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity.