Irreducibility criterion for a finite-dimensional highest weight representation of the sl(2) loop algebra and the dimensions of reducible representations

TL;DR

This paper establishes a precise criterion for when finite-dimensional highest weight representations of the sl(2) loop algebra are irreducible, and provides an algorithm to construct such representations, aiding analysis of related integrable models.

Contribution

It introduces a necessary and sufficient irreducibility criterion and an explicit construction algorithm for highest weight representations of the sl(2) loop algebra.

Findings

Derived an irreducibility criterion for highest weight representations.

Provided an explicit algorithm for constructing representations.

Illustrated the conjecture with specific examples.

Abstract

We present a necessary and sufficient condition for a finite-dimensional highest weight representation of the $sl_2$ loop algebra to be irreducible. In particular, for a highest weight representation with degenerate parameters of the highest weight, we can explicitly determine whether it is irreducible or not. We also present an algorithm for constructing finite-dimensional highest weight representations with a given highest weight. We give a conjecture that all the highest weight representations with the same highest weight can be constructed by the algorithm. For some examples we show the conjecture explicitly. The result should be useful in analyzing the spectra of integrable lattice models related to roots of unity representations of quantum groups, in particular, the spectral degeneracy of the XXZ spin chain at roots of unity associated with the $sl_2$ loop algebra.