Quantization of Drinfel'd doubles

Abstract

Hopf algebra quantizations of 4-dimensional and 6-dimensional real classical Drinfel'd doubles are studied by following a direct "analytic" approach. The full quantization is explicitly obtained for most of the Drinfel'd doubles, except a small number of them for which the dual Lie algebra is either sl(2) or so(3). In the latter cases, the classical r-matrices underlying the Drinfel'd double quantizations contain known standard ones plus additional twists. Several new four and six-dimensional quantum algebras are presented and some general features of the method are emphasized.

23 pages, no figures

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• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• The Drinfeld double gl(n) \oplus t_n
• The Poincaré group as a Drinfel'd double
• Classical Lie algebras and Drinfeld doubles
• From Quantum Universal Enveloping Algebras to Quantum Algebras
• Bases in Lie and Quantum Algebras