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

Robin Lautenbacher, Ralf Köhl

TL;DR

The paper develops and analyzes finite-dimensional spin-like representations of the maximal compact subalgebra $\mathfrak{k}(A)$ of split-real simply-laced Kac-Moody algebras. It constructs and studies generalized spin representations $\mathcal{S}_{\tfrac{1}{2}}, \mathcal{S}_{\tfrac{3}{2}}, \mathcal{S}_{\tfrac{5}{2}}, \mathcal{S}_{\tfrac{7}{2}}$ using an abstract $\Gamma$-matrix framework tied to a normalized 2-cocycle on the root lattice, and proves irreducibility, semisimplicity, and lifting properties to spin covers. A central result is that these higher spin representations lift to the spin cover $Spin(A)$ but not to the maximal compact subgroup $K(A)$, with explicit Weyl-group compatible matrices and natural parametrizations of representation matrices via real roots. The paper also clarifies lift criteria, the role of spin-extended Weyl groups, and the interplay between $W(A)$-action and the higher spin construction, refining earlier results and correcting sign issues in prior work.

Abstract

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

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

TL;DR

The paper develops and analyzes finite-dimensional spin-like representations of the maximal compact subalgebra of split-real simply-laced Kac-Moody algebras. It constructs and studies generalized spin representations using an abstract -matrix framework tied to a normalized 2-cocycle on the root lattice, and proves irreducibility, semisimplicity, and lifting properties to spin covers. A central result is that these higher spin representations lift to the spin cover but not to the maximal compact subgroup , with explicit Weyl-group compatible matrices and natural parametrizations of representation matrices via real roots. The paper also clarifies lift criteria, the role of spin-extended Weyl groups, and the interplay between -action and the higher spin construction, refining earlier results and correcting sign issues in prior work.

Abstract

Given the maximal compact subalgebra of a split-real Kac-Moody algebra of type , we study certain finite-dimensional representations of , that do not lift to the maximal compact subgroup of the minimal Kac-Moody group associated to but only to its spin cover . 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 .
Paper Structure (12 sections, 42 theorems, 174 equations, 2 figures)

This paper contains 12 sections, 42 theorems, 174 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kac-Moody-algebras
  4. The maximal compact subalgebra $\mathfrak{k}(A)$
  5. Higher spin representations
  6. Generalized spin representations and generalized $\Gamma$-matrices
  7. The higher spin representations $\mathcal{S}_{\frac{3}{2}}$ and $\mathcal{S}_{\frac{5}{2}}$
  8. The higher spin representation $\mathcal{S}_{\frac{7}{2}}$
  9. Reducibility and Irreducibility
  10. Lift to $Spin(A)$ and compatibility with action of $W^{spin}(A)$
  11. Maximal compact subgroups, spin covers and a lift criterion
  12. Lifting and interaction with spin-extended Weyl group

Key Result

Proposition 2.6

On $\mathfrak{g}\left(A\right)\left(\mathbb{K}\right)$ there exists an involutive automorphism $\omega$ called the Chevalley involution that is determined by It satisfies $\omega\left(\mathfrak{g}_{\alpha}\right)=\mathfrak{g}_{-\alpha}$ for all $\alpha\in\Delta$.

Figures (2)

  • Figure 1: Two important commutative diagrams for a spherical subdiagram $J$. $\exp_0$ denotes the abstract exponential map of a finite-dimensional Lie algebra to the identity component of a Lie group with that Lie algebra.
  • Figure 2: A commutative diagram for $J$ spherical and $U$ a f.d. module.

Theorems & Definitions (101)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6: This is thm. 1.2 and sec. 1.3 of Kac:
  • Proposition 2.7: Cp. Kac:, thm. 2.2
  • Definition 2.8
  • Definition 2.10
  • Theorem 2.11: Cp. thm. 1.8 of Hainke, originally due to Berman
  • Corollary 2.12: Cp. Berman, thm. 1.31 or Hainke, thm. 1.8
  • ...and 91 more