What question did this study set out to answer?

The research aims to explore finite-dimensional representations of the maximal compact subalgebra k(A) that are associated with the Kac–Moody algebra g(A).

February 8, 2026

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

Key Points

The research aims to explore finite-dimensional representations of the maximal compact subalgebra k(A) that are associated with the Kac–Moody algebra g(A).
Studied representations of the maximal compact subalgebra k(A) of Kac–Moody algebra g(A).
Examined irreducibility and semisimplicity of spin representations.
Analyzed the relationship between these representations and the spin-extended Weyl group.
Identified four known elementary spin representations of k(A).
Demonstrated that these representations do not lift to the maximal compact subgroup K(A).
Derived a partial parametrization of representation matrices using real roots of g(A).

Abstract

Given the maximal compact subalgebra k(A) of a split-real Kac–Moody algebra g(A) of type A, we study certain finite-dimensional representations of k(A) that do not lift to the maximal compact subgroup K(A) of the minimal Kac–Moody group G(A) associated to g(A) but only to its spin cover Spin(A) described in 11. Currently, four elementary of these so-called spin representations are known. We study their (ir)reducibility, semisimplicity, 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 g(A).

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

Key Points

Abstract

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