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).
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Klimyk, A.U. (2001). Nonstandard Quantization of the Enveloping Algebra U(so(n)) and its Applications. In: Duplij, S., Wess, J. (eds) Noncommutative Structures in Mathematics and Physics. NATO Science Series, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0836-5_27
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DOI: https://doi.org/10.1007/978-94-010-0836-5_27
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