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Reduction of symmetries of Dirac structures. (English) Zbl 1252.53094

Summary: We consider a Dirac structure \(D\) on a manifold \(Q\) invariant under a proper action of a Lie group \(G\) on \(Q\). Our aim is to describe the structure of the orbit space \(D/G\) in terms of the structure of \(Q/G\).
If the action of \(G\) on \(Q\) is free, then \(Q\) is a left principal fibre bundle with structure group \(G\) and base manifold \(Q/G\). We denote by \({Q[\mathfrak{g}]}\) the adjoint bundle and by \({Q[\mathfrak{g}^{*}]}\) the coadjoint bundle of \(Q\). We show that, for a free and proper action, \(D/G\) is a maximal isotropic subbundle of the direct sum \({Q[\mathfrak{g}] \oplus Q[\mathfrak{g}^{*}] \oplus T(Q/G) \oplus T^{*}(Q/G)}\).
If the action of \(G\) on \(Q\) is proper but not free, then the orbit space \(Q/G\) is stratified. For each stratum of \(Q/G\), the restriction of \(D/G\) to the stratum can be described in the same way as for a free and proper action.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
57R55 Differentiable structures in differential topology
58A35 Stratified sets
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