A Unified Approach to Poisson Reduction |
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Authors: | Mackenzie K C H |
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Institution: | (1) Department of Pure Mathematics, University of Sheffield, Sheffield, S3 7RH, United Kingdom |
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Abstract: | Given any Poisson action G×P P of a Poisson–Lie group G we construct an object =T
*G*T*
P which has both a Lie groupoid structure and a Lie algebroid structure and which is a half-integrated form of the matched pair of Lie algebroids which J.-H. Lu associated to a Poisson action in her development of Drinfeld's classification of Poisson homogeneous spaces. We use to give a general reduction procedure for Poisson group actions, which applies in cases where a moment map in the usual sense does not exist. The same method may be applied to actions of symplectic groupoids and, most generally, to actions of Poisson groupoids. |
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Keywords: | Poisson– Lie groups Poisson actions Drinfel'd doubles Poisson homogeneous spaces Lie algebroids symplectic groupoids Poisson reduction |
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