Nonstandard Deformation U′ _q (so _n ): The Imbedding U′ _q (so _n ) ⊂ U _q (sl _n ) and Representations

Abstract

Quantum orthogonal groups, quantum Lorentz group and their corresponding quantum algebras are of special interest for modern physics [1–4]. M. Jimbo [5] and V. Drinfeld [6] defined q-deformations (quantum algebras) U _q (g)for all simple complex Lie algebras gby means of Cartan subalgebras and root subspaces. Reshetikhin, Takhtajan and Faddeev [7] defined quantum algebras U _q (g)in terms of the universal R-matrix. However, none of these approaches is able to give a satisfactory presentation of the quantum algebra U _q (so(n,ℂ)) from a viewpoint of some problems in quantum physics and representation theory. In fact, several important problems of theoretical physics demand to define an action of the quantum Lorentz group SO _q (n,1) and of the corresponding quantum algebra (the real forms of the quantum group SO _q (n+ 1, ℂ) and of the quantum algebra U _q (so(n+ 1, ℂ)) respectively) upon the quantum (n + 1)-dimensional linear space. The approaches mentioned above are not satisfactory for such definition. Besides, when considering representations of the quantum groups SO _q (n +1) and SO _q (n,1) we are interested in reducing them onto the quantum subgroup SO _q (n). This reduction would give the analog of the Gel’fand-Tsetlin basis for these representations. However, definitions of quantum algebras mentioned above do not allow the inclusions SO _q (n+ 1) ⊃SO _q (n)and U _q (so(n,l)) ⊃U _q (so(n)). To be able to exploit such reductions we have to consider q-deformations of the Lie algebra so(n+ 1, ℂ) defined in terms of the generators I _k,k−1= E _k,k−1−E _k−1,k (where E _is is the matrix with elements (E _is ) _rt = δ_irδ _st ) rather than by means of Cartan subalgebras and root elements.