Partial Poisson structure on Convenient Fernand Pelletier - PowerPoint PPT Presentation

Partial Poisson structure on Convenient manifolds Partial Poisson structure on Convenient Fernand Pelletier manifolds 0. Outline 1. Introduction Fernand Pelletier 2. Partial Poisson structure LAMA, UMR 5127, CNRS Université de Savoie Mont Blanc 3. Some examples of XXXV Workshop on Geometric Methods in Physics partial Poisson 29 June 2016 manifolds 4. Partial Banach Poisson manifolds and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial

0. Outline Partial 1. Introduction . Poisson structure on 2. Partial Poisson structure. Convenient manifolds 3. Some examples of partial Poisson manifolds. Fernand 4. Banach partial Poisson manifolds and Banach Lie algebroid. Pelletier 5. Partial Poisson, integrability and perspectives. 0. Outline 1. Introduction 2. Partial Poisson structure 3. Some examples of partial Poisson manifolds 4. Partial Banach Poisson manifolds and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial

1. Introduction Partial The concept of Poisson structure is a fundamental Poisson structure on mathematical tool in Mathematical Physic and classical Convenient manifolds Mechanic (specially in finite dimension context ) and, in an Fernand infinite dimensional context, in hydrodynamic framework, in Pelletier mechanism for integrating some evolutionary PDE (for example 0. Outline Kdv), quantum mechanic.... In any of these situations, we have 1. an algebra A of smooth functions on some manifold M Introduction 2. Partial (eventually infinite dimensional) which is provided with a Lie Poisson bracket { , } which satisfies the Leibniz property (called a structure 3. Some Poisson bracket ) and to the derivation g �→ { f, g } in A we examples of partial can associate a vector field X f on M called a the Hamiltonian Poisson manifolds vector field of f . 4. Partial Banach Poisson manifolds and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial

1 : Continuation Partial In infinite dimension, when M is a Banach manifold and Poisson structure on A = C ∞ ( M ) such a framework was firstly defined and studied in a Convenient series of papers by A. Odzijewicz, T. Ratiu and their collaborators manifolds (2003-2009) (see for instance [6]) and we will see how this context is Fernand Pelletier included in our presentation. A more recent, approach was also proposed by K.H. Neeb, 0. Outline H. Sahlmann and T. Thiemann ( "Weak Poisson structures" [4]) 1. Introduction when M is a smooth manifold modelled on a l.c.t.v. space : they consider a subalgebra A of C ∞ ( M ) which is provided with a Poisson 2. Partial Poisson bracket and so that the following separation assumption is satisfied : structure 3. Some ∀ x ∈ M, { d x f ( v ) = 0 , ∀ f ∈ A} = ⇒ { v = 0 } . examples of partial Poisson This condition implies that the Hamiltonian field X f is defined for manifolds any f ∈ A . 4. Partial Banach Poisson manifolds and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial

1 : Continuation Partial Our purpose is to propose, in an infinite dimensional context, a Poisson structure on "Poisson framework" for which the Poisson bracket can be Convenient manifolds defined for some typical local or global smooth functions on Fernand M . Essentially we consider Pelletier the algebra A ( M ) of smooth functions on M whose 0. Outline differential induces a section of a subbundle of T ′ M of 1. T ∗ M Introduction 2. Partial a bundle morphism P : T ′ M → TM such that : Poisson structure { f, g } P = dg ( P ( d f )) defines a Poisson bracket on A . 3. Some examples of Note that under the assumptions of "weak Poisson structures" partial Poisson the vector spaces ∆ x generates by { d f ( x ) , : f ∈ A} does not manifolds give rise to a subbundle of T ∗ M in general. However we have a 4. Partial Banach well defined linear map P x : ∆ x → T x M such that Poisson manifolds P x ( d f ( x )) = X f ( x ) and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial

2. Partial Poisson structure Partial We consider the Kriegl & Michor’s convenient setting ([3]). For short, Poisson structure on a convenient vector space E is a locally convex topological vector Convenient manifolds space (l.c.t.v.s) such that a curve c : R − → E is smooth if and only if Fernand λ ◦ c is smooth for all continuous linear functionals λ on E . We get a Pelletier second topology on E which is the final topology defined by the set 0. Outline of all smooth curves and called the c ∞ -topology. This last topology 1. may be different from the l.c.t.v.s topology and for the c ∞ -topology Introduction 2. Partial E can be not a topological vector space. However Banach spaces and Poisson structure Fréchet spaces are convenient spaces and these two topologies 3. Some coincide. A map f : E → R is smooth if and only if f ◦ c : R → R is examples of partial a smooth map for any smooth curve c in E . Therefore we have an Poisson evident notion of convenient manifold modeled on the c ∞ -topology manifolds 4. Partial of a convenient space. Banach Poisson manifolds and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial

2 : Continuation Partial Now let M be a convenient manifold modeled on convenient Poisson structure on space M . We denote by : p M : TM → M its tangent bundle Convenient manifolds and by p ∗ M : T ∗ M → M its cotangent bundle. Fernand Consider Pelletier a vector subbundle p ′ : T ′ M → M of p ∗ M : T ∗ M → M 0. Outline such that p ′ : T ′ M → M is a convenient bundle 1. Introduction a bundle morphism P : T ′ M → TM which is 2. Partial skew-symmetric i.e. Poisson structure 3. Some < ξ, P ( η ) > = − < η, P ( ξ ) > examples of partial Poisson for all sections ξ and η of T ′ M , where < , > is the bilinear manifolds 4. Partial crossing between T ∗ M and TM . Banach Poisson manifolds and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial

2 : Continuation Partial If ι : T ′ M → T ∗ M is the canonical injection, let A ( M ) be the Poisson structure on set of smooth functions f : M → R such that d f ◦ ι is a section Convenient of p ′ : T ′ M → M . manifolds Fernand So A ( M ) is a sub-algebra of the algebra C ∞ ( M ) of smooth Pelletier functions on M . On A ( M ) we define : 0. Outline 1. { f, g } P = < dg, P ( d f ) > (1) Introduction 2. Partial In these conditions, the relation (1) defines a skew-symmetric Poisson structure bilinear map { , } P : A ( M ) × A ( M ) → C ∞ ( M ) . 3. Some examples of partial Poisson manifolds 4. Partial Banach Poisson manifolds and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial

2 : Continuation Partial In fact the bilinear map { , } P takes values in A ( M ) and satisfies the Poisson structure on Leibniz property { f, gh } P = g { f, h } P + h { f, g } P . Convenient manifolds Fernand Definition Pelletier 0. Outline Let p ′ : T ′ M → M be a convenient subbundle of p ∗ M : T ∗ M → M 1. and P : T ′ M → TM a skew-symmetric morphism. Introduction We say that ( M, A ( M ) , { , } P ) is a partial Poisson structure on 2. Partial Poisson M or ( M, A ( M ) , { , } P ) is a partial Poisson manifold if structure 3. Some the bracket { , } P satisfies the Jacobi identity : examples of partial Poisson { f, { g, h } P } P + { g, { h, f } P } P + { h, { f, g } P } P = 0; manifolds 4. Partial Banach Poisson manifolds and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial

2 : Continuation Partial If M is a Hilbert (resp. Banach, resp. Fréchet) manifold and if Poisson structure on the subbundle T ′ M is a Hilbert (resp. Banach, resp. Fréchet Convenient manifolds bundle), the partial Poisson manifold ( M, A ( M ) , { , } P ) will be Fernand called a partial Poisson Hilbert (resp. Banach, resp. Fréchet) Pelletier manifold . 0. Outline From now the morphism P is fixed we simply denote by 1. { , } the Poisson bracket { , } P . As classically, given a Introduction 2. Partial partial Poisson manifold ( M, A ( M ) , { , } ) , any function Poisson f ∈ A ( M ) is called a Hamiltonian , the associated vector field structure 3. Some X f = P ( d f ) is called a Hamiltonian vector field . In particular examples of partial we have { f, g } = X f ( g ) . Also we have [ X f , X g ] = X { f,g } Poisson manifolds which is equivalent to P ( d { f, g } ) = [ P ( d f ) , P ( dg )] 4. Partial Banach Poisson manifolds and Banach Lie algebroid Fernand Pelletier Partial Poisson structure on Convenient manifolds 5. Partial