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Nonstandard Deformation U′q(son): The Imbedding U′q(son) ⊂ Uq(sln) and Representations

  • A. M. Gavrilik
  • N. Z. Iorgov
  • A. U. Klimyk

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−1E k−1,k (where E is is the matrix with elements (E is ) rt = δirδ st ) rather than by means of Cartan subalgebras and root elements.

Keywords

Irreducible Representation Hopf Algebra Quantum Group Quantum Algebra Dimensional Linear Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • A. M. Gavrilik
    • 1
  • N. Z. Iorgov
    • 1
  • A. U. Klimyk
    • 1
  1. 1.Bogolibov Institute for Theoretical PhysicsUkrainian National Academy of SciencesKievUkraine

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