Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras

TL;DR

This paper investigates certain finite-dimensional spin representations of the maximal compact subalgebra of split-real Kac-Moody algebras, analyzing their properties and how they relate to the algebra's root structure and group extensions.

Contribution

It introduces new insights into the structure and classification of spin representations of maximal compact subalgebras of Kac-Moody algebras, including their reducibility and lift properties.

Findings

Analysis of irreducibility and semi-simplicity of spin representations

Partial parametrization of representation matrices by real roots

Interaction with spin-extended Weyl-group elucidated

Abstract

Given the maximal compact subalgebra $\mathfrak{k}(A)$ of a split-real Kac-Moody algebra $\mathfrak{g}(A)$ of type $A$, we study certain finite-dimensional representations of $\mathfrak{k}(A)$, that do not lift to the maximal compact subgroup $K(A)$ of the minimal Kac-Moody group $G(A)$ associated to $\mathfrak{g}(A)$ but only to its spin cover $Spin(A)$. Currently, four elementary of these so-called spin representations are known. We study their (ir-)reducibility, semi-simplicity, and lift to the group level. The interaction of these representations with the spin-extended Weyl-group is used to derive a partial parametrization result of the representation matrices by the real roots of $\mathfrak{g}(A)$.