Difference between revisions of "Poisson manifold"
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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'''. Examples of Poisson manifolds | + | The pair $(M,\{~,~\})$ is called a '''Poisson manifold'''. A '''Poisson map''' is 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. |
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| + | == Examples of Poisson manifolds == | ||
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| + | Examples of Poisson manifolds include [[Symplectic manifold|symplectic manifolds]] and [[Poisson Lie group|Poisson Lie groups]]. | ||
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| + | 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: | ||
| + | $$ \{f,g\}(v):=\langle [\textrm{d}_v f, \textrm{d}_v], v\rangle. $$ | ||
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| + | == Hamiltonian Systems == | ||
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 11:27, 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. A Poisson map is 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.
Examples of Poisson manifolds
Examples of Poisson manifolds include symplectic manifolds and Poisson Lie groups.
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: $$ \{f,g\}(v):=\langle [\textrm{d}_v f, \textrm{d}_v], v\rangle. $$
Hamiltonian Systems
On a Poisson manifold $(M,\{~,~\})$, any smooth function $h\in C^\infty(M)$ determines a hamiltonian vector field $X_h$ by setting: $$ X_h(f):=\{h,f\}.$$
Poisson manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_manifold&oldid=19522