Classification theorem on irreducible representations of the q-deformed algebra U′q(son)

Abstract

The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandard q-deformation U′q(son) (which does not coincide with the Drinfel'd-Jimbo quantum algebra Uq(son)) of the universal enveloping algebra U(son(ℂ)) of the Lie algebra son(ℂ) when q is not a root of unity. These representations are exhausted by irreducible representations of the classical type and of the nonclassical type. The theorem on complete reducibility of finite-dimensional representations of U′q(son) is proved.