CLASSIFICATION OF IRREDUCIBLE REPRESENTATIONS OF THE q-DEFORMED ALGEBRA U ′ q(son)

A classification of finite dimensional irreducible representations of the nonstandard q-deformation U ′ q (so n) of the universal enveloping algebra U (so(n, C)) of the Lie algebra so(n, C) (which does not coincides with the Drinfeld–Jimbo quantized universal enveloping algebra U q (so n)) is given for the case when q is not a root of unity. It is shown that such representations are exhausted by representations of the classical and nonclassical types. Examples of the algebras U ′ q (so 3) and U ′ q (so 4) are considered in detail. The notions of weights, highest weights, highest weight vectors are introduced. Raising and lowering operators for irreducible finite dimensional representations of U ′ q (so n) and explicit formulas for them are given. They depend on a weight upon which they act. Sketch of proofs of the main assertions are given.