## 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
Full Text:

### References:

 [1] Blankenstein, G.; van der Schaft, A. J., Symmetry and reduction in implicit generalized Hamiltonian systems, Rep. Math. Phys., 47, 57-100, (2001) · Zbl 0978.37046 [2] O. Brahic, R.L. Fernandes, Integration of coupling Dirac structures (in preparation). · Zbl 1327.53106 [3] O. Brahic, R.L. Fernandes, Poisson fibrations and fibered symplectic groupoids, in: Proceedings of the Conference on Poisson Geometry in Mathematics and Physics, Tokyo 2006, AMS Contemporary Mathematics Series. · Zbl 1158.53064 [4] Bursztyn, H.; Cabrera, A., Symmetries and reduction of multiplicative 2-forms, J. Geom. Mech., 4, 2, 111-127, (2012) · Zbl 1260.53131 [5] Bursztyn, H.; Cavalcanti, G.; Gualtieri, M., Reduction of Courant algebroids and generalized complex structures, Adv. Math., 211, 2, 726-765, (2007) · Zbl 1115.53056 [6] Bursztyn, H.; Crainic, M.; Weinstein, A.; Zhu, C., Integration of twisted Dirac brackets, Duke Math. J., 123, 3, 549-607, (2004) · Zbl 1067.58016 [7] Bursztyn, H.; Radko, O., Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier, 53, 1, 309-337, (2003) · Zbl 1026.58019 [8] Courant, T., Dirac manifolds, Trans. Amer. Math. Soc., 319, 2, 631-661, (1990) · Zbl 0850.70212 [9] Crainic, M.; Fernandes, R. L., Integrability of Lie brackets, Ann. of Math. (2), 157, 575-620, (2003) · Zbl 1037.22003 [10] Crainic, M.; Fernandes, R. L., Integrability of Poisson brackets, J. Differential Geom., 66, 71-137, (2004) · Zbl 1066.53131 [11] Crainic, M.; Fernandes, R. L., Lectures on integrability of Lie brackets, Geom. Topol. Monogr., 17, 1-107, (2011) · Zbl 1227.22005 [12] Fernandes, R. L.; Ortega, J. P.; Ratiu, T., Momentum maps in Poisson geometry, Amer. J. Math., 131, 5, 1261-1310, (2009) · Zbl 1180.53083 [13] Iglesias Ponte, D.; Wade, A., Integration of Dirac Jacobi structures, J. Phys. A: Math. Gen., 39, 4181-4190, (2006) · Zbl 1094.70013 [14] Jotz, M.; Ratiu, T., Dirac optimal reduction, Int. Math. Res. Not., 1, 84-155, (2013) · Zbl 1275.37028 [15] Wade, A., Poisson fiber bundles and coupling Dirac structures, Ann. Global Anal. Geom., 33, 207-217, (2008) · Zbl 1151.53070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.