International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 2, Pages 225-262
Classification theorem on irreducible representations of the -deformed algebra
Mathematical Methods in Theoretical Physics Department, Bogolyubov Institute for Theoretical Physics, Metrologichna Street, Kiev 03143, Ukraine
Received 8 August 2004
Copyright © 2005 N. Z. Iorgov and A. U. Klimyk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandard -deformation (which does not coincide with the Drinfel'd-Jimbo quantum algebra ) of the universal enveloping algebra of the Lie algebra when 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 is proved.