Vol. 245, No. 1, 2010

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Quantization of Poisson–Hopf stacks associated with group Lie bialgebras

Gilles Halbout and Xiang Tang

Vol. 245 (2010), No. 1, 99–118
Abstract

Let G be a simply connected Poisson–Lie group and g its Lie bialgebra. Suppose that g is a group Lie bialgebra. This means that there is an action of a discrete group Γ on G deforming the Poisson structure into coboundary equivalent ones. This induces the existence of a Poisson–Hopf algebra structure on the direct sum over Γ of formal functions on G, with Poisson structures translated by Γ. A quantization of this algebra can be obtained by taking the linear dual of a quantization of the Γ Lie bialgebra g, which is the infinitesimal of a Γ Poisson–Lie group. In this paper we find out an interesting structure on the dual Lie group G. We prove that we can construct a stack of Poisson–Hopf algebras and prove the existence of the associated deformation quantization of it. This stack can be viewed as the function algebra on “the formal Poisson group” dual to the original Γ Poisson–Lie group. To quantize this stack, we apply Drinfeld functors to quantization of the associated Γ Lie bialgebra.

Keywords
stack, Poisson, Hopf, Lie bialgebra
Mathematical Subject Classification 2000
Primary: 17B37
Secondary: 58H05
Milestones
Received: 27 October 2008
Revised: 13 August 2009
Accepted: 14 August 2009
Published: 20 January 2010
Authors
Gilles Halbout
Institut de Mathématiques et de Modélisation de Montpellier
Université de Montpellier 2
CC5149, Place Eugène Bataillon
F-34095 Montpellier CEDEX 5
France
Xiang Tang
Department of Mathematics
Washington University
St. Louis, MO 63130
United States