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Quantum coadjoint action. (English) Zbl 0747.17018

This is a continuation of an earlier paper by the first two authors [Proc. Colloq. in Honour of J. Dixmier, Paris 1989, Prog. Math. 92, 471–506 (1990; Zbl 0738.17008)] on representations of quantum groups at roots of 1. Solutions are given to most of the (suitably modified) conjectures of this earlier paper concerning the center and the quantum coadjoint action. Here the quantum groups are simply connected, the adjoint quantum group being the subalgebra of invariants of the center of the corresponding simply connected Lie group.
From the authors’ introduction: “The classical orbit method relates representations of a Lie group to the orbits of the coadjoint action of this group in the dual of the Lie algebra. The basic observation of the present paper is that representations of a quantum group of roots of 1 are closely related to the orbits of the action of the corresponding group on itself by conjugation.”
In a final section, the authors interpret their results in the language of Poisson algebraic groups.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16T20 Ring-theoretic aspects of quantum groups

Citations:

Zbl 0738.17008
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References:

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