Polarized Poisson structures
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Polarized Poisson structures Azzouz Awane1 · Benali Ismail1 · El Amine Souhaila1 Received: 15 July 2020 / Accepted: 5 October 2020 © The Managing Editors 2020
Abstract We study various properties of polarized Poisson structure subordinate to the polarized symplectic structure.We study also the notion of a polarized Poisson manifold, i.e., a Poisson manifold foliated by coisotropic submanifolds. We show that the characteristic distribution of a polarized Poisson structure is completely integrable and its leaves are symplectic. In the particular case when the foliation is Lagrangian, we show that, the polarized Poisson manifold is also foliated by polarized symplectic leaves, and we prove Darboux’s theorem corresponding to a Lagrangian foliation with respect to a polarized Poisson manifold. Keywords Foliation · Symplectic structures · Hamiltonian vector fields · Poisson manifold · Foliated Poisson manifold Mathematics Subject Classification 53C12 · 53D05 · 37K05 · 53D17
1 Introduction Let (M, θ ) be a symplectic manifold of dimension 2n. A real polarization F of (M, θ ) is a Lagrangian foliation, i.e. an n-codimensional foliation F whose leaves are Lagrangian submanifolds of (M, θ ). The triple (M, θ, F) is called a polarized symplectic manifold. The notion of a polarization plays an important role in the geometric quantization theory of Kostant - Souriau, see for example, Woodhouse (1980). Concerning the geometry of Lagrangian foliations, interesting properties have been highlighted by Weinstein (1976), Dazord (1981) , Molino (1984), etc.
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Azzouz Awane [email protected] Benali Ismail [email protected] El Amine Souhaila [email protected]
1
LAMS, Faculty of Sciences of Ben M’sik, Hassan II University of Casablanca, Bd Driss Harti, B.P.7955, Casablanca, Morocco
123
Beitr Algebra Geom
The leaves of the real polarization F are affine submanifolds (Dazord 1981). Darboux’s theorem of polarized symplectic manifolds shows that around each point x0 of M, there is a local coordinate system (xi , yi )1≤i≤n defined on an open U of M, such that n d xi ∧ dyi and, F|U is defined by the equations : dy1 = 0, . . . , dyn = 0. θ|U = i=1 A polarized Hamiltonian vector field is a foliate vector field X such that the Pfaffian form i (X ) θ is exact. When a smooth real function H on M satisfies the relation i (X ) θ = −d H , we say that H is a polarized Hamiltonian mapping. In particular, the Lagrangian foliation F is affine and each polarized Hamiltonian mapping H is locally affine function on each leaf of this foliation. Locally, with respect to Darboux’s coordinates, the polarized Hamiltonian H takes the following form H=
n
a j (y1 , . . . , yn )x j + b(y1 , . . . , yn ),
j=1
where a j and b are basic functions. The set of all polarized Hamiltonian mappings, denoted by H (M, F), is a proper linear subspace of C ∞ (M). The restriction of the natural bracket {, } on C ∞ (M) defined by the symplectic form θ induces a natural law of Lie algebra on H (M, F), denoted also by {, } satisfies in addition, th
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