Difference between revisions of "Poisson manifold"
From Encyclopedia of Mathematics
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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: | 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: | ||
$$ \{f,gh\}=\{f,g\}h+g\{f,h\},\qquad \forall f,g,h\in C^\infty(M).$$ | $$ \{f,gh\}=\{f,g\}h+g\{f,h\},\qquad \forall f,g,h\in C^\infty(M).$$ | ||
| − | The pair $(M,\{~,~\})$ is called a '''Poisson manifold'''. | + | The pair $(M,\{~,~\})$ is called a '''Poisson manifold'''. Examples of Poisson manifolds inlcude [[Symplectic manifold|symplectic manifolds]] and [[Poisson Lie group|Poisson Lie groups]]. |
| − | On a Poisson manifold $(M,\{~,~\}), any smooth function $h\in C^\infty(M)$ determines a '''hamiltonian vector field''' $X_h$ by setting: | + | On a Poisson manifold $(M,\{~,~\})$, any smooth function $h\in C^\infty(M)$ determines a '''hamiltonian vector field''' $X_h$ by setting: |
$$ X_h(f):=\{h,f\}.$$ | $$ X_h(f):=\{h,f\}.$$ | ||
Revision as of 09:54, 30 August 2011
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. Examples of Poisson manifolds inlcude symplectic manifolds and Poisson Lie groups.
On a Poisson manifold $(M,\{~,~\})$, any smooth function $h\in C^\infty(M)$ determines a hamiltonian vector field $X_h$ by setting: $$ X_h(f):=\{h,f\}.$$
How to Cite This Entry:
Poisson manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_manifold&oldid=19521
Poisson manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_manifold&oldid=19521