Generalized spin representations. Part 2: Cartan-Bott periodicity for the split real En series

Abstract

In this article we analyze the quotients of the maximal compact subalgebras of the split real Kac-Moody algebras of the En series resulting from the generalized spin representations introduced in part 1. It turns out that these quotients satisfy a Cartan-Bott periodicity. Our findings are also meaningful in the finite-dimensional cases of A2 + A1, A4, D5, E6, E7, E8, where it turns out that the generalized spin representation is injective. Consequently the observed Cartan-Bott periodicity provides a structural explanation for the seemingly sporadic isomorphism types of the maximal compact Lie subalgebras of the split real Lie algebras of types E6, E7, E8.