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Abstract
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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.
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Keywords
stack, Poisson, Hopf, Lie bialgebra
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Mathematical Subject Classification 2000
Primary: 17B37
Secondary: 58H05
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Milestones
Received: 27 October 2008
Revised: 13 August 2009
Accepted: 14 August 2009
Published: 20 January 2010
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