Representations of Involutory Subalgebras of Affine Kac–Moody Algebras

Abstract

AbstractWe consider the subalgebras of split real, non-twisted affine Kac–Moody Lie algebras that are fixed by the Cartan–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 $${\mathbb {N}}$$ 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 and work out a detailed example in the case of the affine extension of $${\mathfrak {e}}_8$$ e 8 .