Generalized Drinfeld polynomials for highest weight vectors of the Borel subalgebra of the $sl_2$ loop algebra

Abstract

In a Borel subalgebra U(B) of the sl(2) loop algebra, we introduce a highest weight vector $\Psi$. We call such a representation of U(B) that is generated by $\Psi$ highest weight. We define a generalization of the Drinfeld polynomial for a finite-dimensional highest weight representation of U(B). We show that every finite-dimensional highest weight representation of the Borel subalgebra is irreducible if the evaluation parameters are distinct. We also discuss the necessary and sufficient conditions for a finite-dimensional highest weight representation of U(B) to be irreducible.