Classification of irreducible representations of the q-deformed algebra U'_q(so_n)

Details Classification of irreducible representations of the q-deformed algebra U'_q(so_n) Classification of irreducible representations of the q-deformed algebra U'_q(so_n)

2001-10-03

arxiv.org/abs/math/0110038v1

Mathematics

Quantum Algebra

13 pages, LaTeX. Report on Int. Conf. on Functional Analysis (Kiev, August 2001)

Scientific paper

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.

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