Groupoïdes symplectiques doubles des groupes de Lie-Poisson. (Symplectic double groupoids of Poisson Lie groups). (French) Zbl 0701.58025

In previous work [A. Coste, P. Dazord, and A. Weinstein, Publ. Dép. Math., Nouv. Sér., Univ. Claude-Bernard, Lyon 2/A, 1-62 (1987; Zbl 0668.58017) and the second author, Bull. Am. Math. Soc., New Ser. 16, 101-104 (1987; Zbl 0618.58020)], one of the present authors has shown that Poisson manifolds may be regarded as infinitesimal invariants for symplectic groupoids, as Lie algebras are for Lie groups. Firstly, the bracket of functions on a Poisson manifold P can be lifted to a bracket of 1-forms, and this bracket makes the cotangent bundle \(T^*P\) into a Lie algebroid. Secondly, a groupoid \(\Gamma\) induces a Poisson structure on its space of units \(\Gamma_ 0\), and the Lie algebroid structure on \(T^*\Gamma_ 0\) induced by the Poisson structure identifies canonically with the Lie algebroid associated to the groupoid structure on \(\Gamma\). Not all abstractly given Poisson structures arise from symplectic groupoids in this way, but for those that do, the groupoid manifold provides a full realization for the Poisson structure [in the sense of A. Weinstein, J. Differ. Geom. 18, 523-557 (1983); Zbl 0524.58011)].
The present paper is concerned with the integrability of Poisson Lie groups; that is, with Poisson structures on group manifolds which make the multiplication a Poisson map (“grouped” in the terminology of V. G. Drin’feld, Sov. Math., Dokl. 27, 68-71 (1983); translation from Dokl. Akad. Nauk SSSR 268, 285-287 (1983; Zbl 0526.58017)]). In this case one seeks not merely a symplectic groupoid integrating the cotangent Lie algebroid, but a symplectic groupoid with additional symmetry, compatible with the group structure on the space of units. The authors show that in fact the cotangent Lie algebroid of a connected Poisson Lie group G integrates to a symplectic groupoid \(\Gamma\) (with space of units G), which admits a second symplectic groupoid structure whose space of units is \(G^*\), the dual group of G; this \(\Gamma\) is a symplectic double groupoid in a natural sense, and embodies the duality properties of the Poisson Lie group.
In a paper written earlier [J. Differ. Geom. 31, No.2, 501-526 (1990; Zbl 0673.58018)], the authors showed that the Lie bialgebra of a Poisson Lie group is a particular example of a double Lie algebra, and that a double Lie algebra may, under suitable connectivity and other conditions, be integrated to a double Lie group [compare also S. Majid, Pac. J. Math. 141, No.2, 311-332 (1990). Double Lie groups in this sense are double groupoids (this is not quite as obvious as the terminology suggests), but a significant distinction between the main result of this paper and the results for double Lie algebras is that the symplectic double groupoid found here need not be of the type which arises from a double Lie group.
Reviewer: K.Mackenzie


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58H05 Pseudogroups and differentiable groupoids
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)