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

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.