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We present a necessary and sufficient condition for a finite-dimensionalhighest weight representation of the $sl_2$ loop algebra to be irreducible. Inparticular, for a highest weight representation with degenerate parameters ofthe highest weight, we can explicitly determine whether it is irreducible ornot. We also present an algorithm for constructing finite-dimensional highestweight representations with a given highest weight. We give a conjecture thatall the highest weight representations with the same highest weight can beconstructed by the algorithm. For some examples we show the conjectureexplicitly. The result should be useful in analyzing the spectra of integrablelattice models related to roots of unity representations of quantum groups, inparticular, the spectral degeneracy of the XXZ spin chain at roots of unityassociated with the $sl_2$ loop algebra.Comment: 32 pages with no figure; with corrections on the published versio

journal of statistical mechanics

Irreducibility criterion for a finite-dimensional highest weight representation of the<i>sl</i><sub>2</sub>loop algebra and the dimensions of reducible representations

Irreducibility criterion for a finite-dimensional highest weight representation of thesl2loop algebra and the dimensions of reducible representations

Irreducibility criterion for a finite-dimensional highest weight representation of thesl₂loop algebra and the dimensions of reducible representations