Branching rules for winding subalgebras of the affine Kac--Moody algebras $A^{(1)}_1$ and $A^{(2)}_2$

TL;DR

This paper explores how integrable highest-weight representations of affine Kac--Moody algebras decompose when restricted to winding subalgebras, revealing closure properties and explicit results for specific types.

Contribution

It extends finite-dimensional branching results to affine Kac--Moody algebras, analyzing the structure of restrictions to winding subalgebras and providing explicit decompositions for certain types.

Findings

Restriction decomposes into irreducible components with finite multiplicities.

The set of highest weights is closed under addition.

Explicit results are provided for types A^{(1)}_1 and A^{(2)}_2.

Abstract

We study branching problems for affine Kac--Moody algebras. Unlike the finite-dimensional case, an affine Kac--Moody algebra may contain proper subalgebras isomorphic to itself, such as winding subalgebras obtained by rescaling the loop parameter. We investigate the restriction of integrable highest-weight representations to such subalgebras. The restriction remains integrable and decomposes into irreducible components with finite multiplicities, encoded by pairs of highest weights. We show that this set is closed under addition, extending a result of Brion and Knop to the affine setting. We also give a partial description of this set and provide explicit results for types $A^{(1)}_1$ and $A^{(2)}_2$.