The six-vertex model at roots of unity and some highest weight representations of the sl(2) loop algebra

TL;DR

This paper explores the structure of highest weight representations of the sl(2) loop algebra in relation to the six-vertex model at roots of unity, providing criteria for irreducibility and examples of reducible cases.

Contribution

It introduces a simple proof for the irreducibility of highest weight representations with distinct evaluation parameters and offers a general criterion for irreducibility.

Findings

Every highest weight representation with distinct evaluation parameters is irreducible.

A criterion for determining irreducibility of highest weight representations.

An example of a reducible indecomposable highest weight representation and its dimension.

Abstract

We discuss irreducible highest weight representations of the sl(2) loop algebra and reducible indecomposable ones in association with the sl(2) loop algebra symmetry of the six-vertex model at roots of unity. We formulate an elementary proof that every highest weight representation with distinct evaluation parameters is irreducible. We present a general criteria for a highest weight representation to be irreducble. We also give an example of a reducible indecomposable highest weight representation and discuss its dimensionality.