Coisotropic calculus and Poisson groupoids.

*(English)*Zbl 0642.58025The natural extension to Poisson manifolds of the calculus of Lagrangian submanifolds (canonical relations) in symplectic manifolds is shown to be a calculus of coisotropic submanifolds. These are submanifolds for which the defining ideal of functions is closed under Poisson bracket. For instance, a map is a Poisson map if and only if its graph is coisotropic when the product of the domain and the range is given the “difference” Poisson structure.

In the first two sections of the paper, basic properties of the calculus are derived and applications are given to the reduction of Poisson manifolds. In the third section, a Lie algebroid structure is shown to exist on the conormal bundle of every coisotropic submanifold. In the final section, a notion of Poisson groupoid is defined which encompasses on the one hand the symplectic groupoids of M. V. Karasev [Izv. Akad. Nauk SSSR, Ser. Mat. 50, No.3, 508-538 (1986; Zbl 0608.58023)] and the author [Bull. Am. Math. Soc., New. Ser. 16, 101-104 (1987; Zbl 0618.58020)] and on the other the Poisson groups of V. G. Drinfel’d [Sov. Math., Dokl. 27, 68-71 (1983); translation from Dokl. Akad. Nauk SSSR 768, 285-287 (1983; Zbl 0526.58017)]. A duality between Poisson groupoids is defined and some examples are given, but the full extension of Drinfel’d’s duality in the group case is only outlined; it is based on the notion of symplectic double groupoid.

In the first two sections of the paper, basic properties of the calculus are derived and applications are given to the reduction of Poisson manifolds. In the third section, a Lie algebroid structure is shown to exist on the conormal bundle of every coisotropic submanifold. In the final section, a notion of Poisson groupoid is defined which encompasses on the one hand the symplectic groupoids of M. V. Karasev [Izv. Akad. Nauk SSSR, Ser. Mat. 50, No.3, 508-538 (1986; Zbl 0608.58023)] and the author [Bull. Am. Math. Soc., New. Ser. 16, 101-104 (1987; Zbl 0618.58020)] and on the other the Poisson groups of V. G. Drinfel’d [Sov. Math., Dokl. 27, 68-71 (1983); translation from Dokl. Akad. Nauk SSSR 768, 285-287 (1983; Zbl 0526.58017)]. A duality between Poisson groupoids is defined and some examples are given, but the full extension of Drinfel’d’s duality in the group case is only outlined; it is based on the notion of symplectic double groupoid.

Reviewer: A.Weinstein

##### MSC:

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |