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

TL;DR

This paper explores the structure of maximal compact subalgebras in split real E-series Kac-Moody algebras, revealing a periodicity pattern that explains certain isomorphisms and extends to finite-dimensional cases.

Contribution

It demonstrates Cartan-Bott periodicity in quotients of maximal compact subalgebras of split real E-series Kac-Moody algebras, providing a structural insight into their isomorphism types.

Findings

Quotients satisfy Cartan-Bott periodicity.

Generalized spin representation is injective in finite cases.

Periodic pattern explains sporadic isomorphisms.

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.