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Integrability of Poisson brackets. (English) Zbl 1066.53131

The authors found in a previous paper [Ann. Math. (2) 157, No. 2, 575–620 (2003; Zbl 1037.22003)] the obstructions for integrating Lie algebroids to Lie groupoids. In the present paper they investigate the corresponding obstructions in the case of the cotangent Lie algebroid \(T^*M\) of a Poisson manifold \(M\). The main result is that the Poisson manifold \(M\) is integrable by a symplectic groupoid if and only if the Lie algebroid \(T^*M\) is integrable. This happens if and only if the monodromy groups \({\mathcal N}_x\), with \(x\in M\), are locally uniformly discrete. We mention that, for every \(x\in M\), the corresponding monodromy group \({\mathcal N}_x\) is an additive subgroup of the co-normal vector space \(\nu_x^*(L)\) to the symplectic leaf \(L\) through the point \(x\). (Actually, \(\nu_x^*(L)\) is a Lie algebra and \({\mathcal N}_x\) is contained in its center.) Another important result obtained in the present paper is that a Poisson manifold \(M\) is integrable if and only if it has a complete symplectic realization \(\varphi\colon S\to M\).
The paper also includes a discussion of Poisson brackets induced on submanifolds. In particular, one establishes sufficient conditions for a submanifold of an integrable Poisson manifold to be in turn integrable with respect to the induced bracket. The last section of the paper is devoted to Morita equivalence. Recall, that the Morita equivalence as introduced in the paper by P. Xu [Commun. Math. Phys. 142, No. 3, 493–509 (1991; Zbl 0746.58034)] only applies to integrable Poisson manifolds. In the present paper a weak Morita equivalence is studied in the case of general Poisson manifolds, and a list of invariants with respect to that weak equivalence relation is supplied.
To conclude with, we note that the paper under review is written in a clear style and includes many illuminating examples.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
22A22 Topological groupoids (including differentiable and Lie groupoids)
58H05 Pseudogroups and differentiable groupoids
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties