Irreducibility criterion for a finite-dimensional highest weight representation of thesl2 loop algebra and the dimensions of reducible representations

Abstract

We present a necessary and sufficient condition for a finite-dimensional highest weight representationof the sl2 loop algebra to be irreducible. In particular, for a highest weight representation withdegenerate parameters of the highest weight, we can explicitly determine whether it isirreducible or not. We also present an algorithm for constructing finite-dimensional highestweight representations with a given highest weight. We give a conjecture thatall of the highest weight representations with the same highest weight can beconstructed by the algorithm. For some examples we show the conjecture explicitly.The result should be useful in analysing the spectra of integrable lattice modelsrelated to roots of unity representations of quantum groups, in particular, thespectral degeneracy of the XXZ spin chain at roots of unity associated with thesl2 loop algebra.