Poisson manifold

Poisson manifold

A Poisson bracket on a smooth manifold $M$ is a Lie bracket $\{~,~\}$ on the space of smooth functions $C^\infty(M)$ which, additionally, satisfies the Leibniz identity: $$ \{f,gh\}=\{f,g\}h+g\{f,h\},\qquad \forall f,g,h\in C^\infty(M).$$ The pair $(M,\{~,~\})$ is called a Poisson manifold. A smooth map between Poisson manifolds $\phi:(M,\{~,~\}_M)\to (N,\{~,~\}_N)$ such that the induced pullback map $\phi^*:C^\infty(N)\to C^\infty(M)$ is a Lie algebra morphism is called a Poisson map.

Examples of Poisson manifolds

Examples of Poisson manifolds include symplectic manifolds, linear Poisson structures and Poisson Lie groups.

Symplectic manifolds

If $(S,\omega)$ is any symplectic manifold and $f\in C^\infty(M)$ is a smooth function then one defines a vector field $X_f$ on $S$, called the hamiltonian vector field associated to $f$, by setting $$ i_{X_f}\omega =\mathrm{d}f. $$ The associated Poisson bracket on $S$ is then given by: $$ \{f,g\}(v):=X_f(g)=-X_g(f).$$

Heisenberg Poisson bracket

If $(S,\omega)$ is any symplectic manifold with associated Poisson bracket $\{~,~\}_S$ then one can define a new Poisson bracket on $M:=S\times\mathbb{R}$ by setting: $$ \{f,g\}_M(x,t)=\{f(\cdot,t),g(\cdot,t)\}_S(x).\] This is called the '''Heisenberg Poisson bracket'''. Actually the same construction can be performed replacing $S$ by any Poisson manifold. ==='"`UNIQ--h-4--QINU`"' Linear Poisson brackets === A Poisson bracket on a [[vector space]] $V$ is called a '''linear Poisson bracket''' if the Poisson bracket of any two linear functions is again a linear function. Since linear functions form a vector space $V^*$ this means that a linear Poisson bracket in $V$ determines a [[Lie algebra]] structure on $\mathfrak{g}:=V^*$. Conversely, if $\mathfrak{g}$ is a finite dimensional Lie algebra then its dual vector space $V:=\mathfrak{g}^*$ carries a linear Poisson bracket which is given by the formula: $$ \{f,g\}(v):=\langle [\textrm{d}_v f, \textrm{d}_v], v\rangle. $$ =='"`UNIQ--h-5--QINU`"' Hamiltonian Systems == On a Poisson manifold $(M,\{~,~\})$, any smooth function $h\in C^\infty(M)$ determines a '''hamiltonian vector field''' $X_h$ by setting: $$ X_h(f):=\{h,f\}.$$