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Integrability and reduction of Hamiltonian actions on Dirac manifolds. (English) Zbl 1298.53082

Summary: For a Hamiltonian, proper and free action of a Lie group \(G\) on a Dirac manifold \((M, L)\), with a regular moment map \(\mu : M \to \mathfrak{g}^\ast\), the manifolds \(M / G\), \(\mu^{- 1}(0)\) and \(\mu^{- 1}(0) / G\) all have natural induced Dirac structures. If \((M, L)\) is an integrable Dirac structure, we show that \(M / G\) is always integrable, but \(\mu^{- 1}(0)\) and \(\mu^{- 1}(0) / G\) may fail to be integrable, and we describe the obstructions to their integrability.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
53D20 Momentum maps; symplectic reduction
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