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On the defining relations of quantum superalgebras. (English) Zbl 0736.17015
Summary: The quantum universal enveloping algebra of simple Lie algebras can be presented in terms of generators and defining relations. These relations are deformations of the standard commutation and Serre relations obeyed by the Chevalley generators. In the case of superalgebras, it has been understood [cf. D. Leites and V. Serganova, Proc. Euler IMI Conf. Quantum Groups, Leningrad 1990, Lect. Notes Math. 1510 (1992)] that extra relations need to be added to the Serre-like ones. A deformation of these extra relations has to be provided when quantizing superalgebras. This point has been overlooked so far [cf. M. Chaichan and P. Kulish, Phys. Lett. B 234, 72-80 (1990); R. Floreanini, V. P. Spiridonov and L. Vinet, Commun. Math. Phys. 137, No. 1, 149-160 (1991; Zbl 0735.17012)].
The purpose of this letter is to identify the \(q\)-deformation of the extra defining relations for the quantum superalgebras \(sl_ q(m,n)\) and \(osp_ q(m,2n)\), using their \(q\)-oscillator realizations.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17A70 Superalgebras
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