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

2024/09/11 by Robin Lautenbacher, Lautenbacher, Robin, Ralf Köhl +1
Mathematics · #17B67 #81R10 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2409.07247

openalex publication_date 2024/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given the maximal compact subalgebra \mathfrakk(A) of a split-real Kac-Moody algebra \mathfrakg(A) of type A, we study certain finite-dimensional representations of \mathfrakk(A), that do not lift to the maximal compact subgroup K(A) of the minimal Kac-Moody group G(A) associated to \mathfrakg(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 \mathfrakg(A).

Related

• Generalized spin representations
• Spin covers of maximal compact subgroups of Kac-Moody groups and\n spin-extended Weyl groups
• Tensor factorization and Spin construction for Kac-Moody algebras
• Generalized spin representations. Part 2: Cartan-Bott periodicity for\n the split real En series
• Spin Structures of Kac-Moody Type