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      Bidynamical Poisson Groupoids

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          Abstract

          We give relations between dynamical Poisson groupoids, classical dynamical Yang--Baxter equations and Lie quasi-bialgebras. We show that there is a correspondance between the class of bidynamical Lie quasi-bialgebras and the class of bidynamical Poisson groupoids. We give an explicit, analytical and canonical equivariant solution of the classical dynamical Yang--Baxter equation (classical dynamical \(\ell\)-matrices) when there exists a reductive decomposition \(\g=\l\oplus\m\), and show that any other equivariant solution is formally gauge equivalent to the canonical one. We also describe the dual of the associated Poisson groupoid, and obtain the characterization that a dynamical Poisson groupoid has a dynamical dual if and only if there exists a reductive decomposition \(\g=\l\oplus\m\).

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          Coisotropic calculus and Poisson groupoids

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            The non-commutative Weil algebra

            Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo map extends to a linear map from compactly supported distributions on the Lie algebra g to compactly supported distributions on the Lie group G, which is a ring homomorphism for G-invariant distributions. In this paper we obtain analogues of the Duflo map and of Duflo's theorem in the context of equivariant cohomology of G-manifolds. Our result involves a non-commutative version of the Weil algebra and of the de Rham model of equivariant cohomology.
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              Manin Pairs and Moment Maps

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                Author and article information

                Journal
                11 October 2006
                Article
                math/0610379
                Custom metadata
                17B62; 53D17; 58H05; 35Q99
                LaTeX, 22 pages
                math.QA math-ph math.MP

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