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Inhomogeneous quantum groups and their quantized universal enveloping algebras. (English) Zbl 0779.17017
After recalling the definition of \(SL_ q(N)\) and \(IGL_ q(N)\) [the latter being defined earlier, see the authors, Z. Phys. C 53, 79-82 (1992)], the inhomogenization of \(GL_ q(N)\), it is shown that the commutation relations of this inhomogeneous quantum group can be given in \(R\)-matrix form. The main result of the paper is the construction of the universal enveloping algebra \(U_ q(IGL(N))\) as a certain Hopf subalgebra (algebra of regular functionals) of the dual Hopf algebra of \(IGL_ q(N)\). This is achieved pulling back regular functionals on \(SL_ q(N)\) to \(IGL_ q(N)\) via some Hopf algebra epimorphism \(IGL_ q(N)\to SL_ q(N)\) and introducing suitable extra functionals corresponding to the invertible element and to the translations.
Finally it is shown that the quantum universal enveloping algebra acts as linear operators on the quantum space corresponding to the translations, and that one can obtain the partial derivatives of a known covariant differential calculus on this quantum space as convolution of some regular functionals.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
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