Classification of irreducible representations of the q-deformed algebra U'_q(so_n)
Abstract
Description
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.
13 pages, LaTeX. Report on Int. Conf. on Functional Analysis (Kiev, August 2001)
13 pages, LaTeX. Report on Int. Conf. on Functional Analysis (Kiev, August 2001)