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

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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.