Dirac brackets and reduction of invariant bi-Poisson structures

2016/05/11 by Mykytyuk, Ihor V., Panasyuk, Andriy
#53D17 37J15 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1605.03382

Abstract

Let X be a manifold with a bi-Poisson structure \η^t\ generated by a pair of G-invariant symplectic structures ω₁ and ω₂, where the Lie group G acts properly on X. Let H be some isotropy subgroup for this action representing the principle orbit type and X^r_\mathfrakh be the submanifold of X consisting of the points in X with the stabilizer algebra equal to the Lie algebra \mathfrakh of H and with the stabilizer group conjugated to H in G. We prove that the pair of symplectic structures ω₁|_X^r_\mathfrakh and ω₂|_X^r_\mathfrakh generates an N(H⁰)/H⁰-invariant bi-Poisson structure on X^r_\mathfrakh, where N(H⁰) is the normalizer in G of the identity component H⁰ of H. The action of \widetilde G=N(H⁰)/H⁰ on X^r_\mathfrakh is locally free and proper and, moreover, the spaces A^G of G-invariant functions on X and A^{\widetilde G} of \widetilde G-invariant functions on X^r_\mathfrakh can be canonically identified and therefore the bi-Poisson structure \(η^t)'\ induced on A^G≃ A^{\widetilde G} can be treated as the reduction with respect to a \em locally free action of a Lie group which essentially simplifies the study of \(η^t)'\.

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