Nonstandard Quantization of the Enveloping Algebra U(so(n)) and its Applications

Abstract

Quantum orthogonal groups, quantum Lorentz groups and their corresponding quantum algebras are of special interest for modern mathematical physics (see, for example, [1] and [2]). M. Jimbo [3] and V. Drinfeld [4] defined g-deformations (quantum algebras) Uq(g) for all simple complex Lie algebras g by means of Cartan subalgebras and root subspaces (see also [5] and [6]). Reshetikhin, Takhtajan and Faddeev [7] defined quantum algebras Uq(g) in terms of the quantum R- matrix satisfying the quantum Yang-Baxter equation. However, these approaches do not give a satisfactory presentation of the quantum algebra Uq(so(n, (C)) from a viewpoint of some problems in quantum physics and representation theory. When considering representations of the quantum groups SOq(n + 1) and SOq(n, 1) we are interested in reducing them onto the quantum subgroup SOq(n). This reduction would give an analogue of the Gel’fand-Tsetlin basis for these representations. However, definitions of quantum algebras mentioned above do not allow the inclusions Uq(so(n + 1,C)) D Uq(so(n,C)) and Uq(sonii) D Uq(son). To be able to exploit such reductions we have to consider g-deformations of the Lie algebra so(n +1, C) defined in terms of the generators Ik,k-i = Ek^k-i — Ek-i,k (where Eis is the matrix with elements (Eis)rt = δirδst) rather than by means of Cartan subalgebras and root elements. To construct such deformations we have to deform trilinear relations for elements Ik,k-1 instead of Serre’s relations (used in the case of Jimbo’s quantum algebras). As a result, we obtain the associative algebra which will be denoted as Uq(so(n, (ℂ)). This g-deformation was first constructed in [8]. It permits one to construct the reductions of U’q(so(n + 1, ℂ) onto U’q(so(n,ℂ).