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Difference between revisions of "Poisson manifold"

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== Examples of Poisson manifolds ==
 
== Examples of Poisson manifolds ==
  
Examples of Poisson manifolds include [[Symplectic manifold|symplectic manifolds]] and [[Poisson Lie group|Poisson Lie groups]].  
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Examples of Poisson manifolds include [[Symplectic manifold|symplectic manifolds]], linear Poisson structures and [[Poisson Lie group|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:
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=== Symplectic manifolds ===
$$ \{f,g\}(v):=\langle [\textrm{d}_v f, \textrm{d}_v], v\rangle. $$  
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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
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$$ i_{X_f}\omega =\mathrm{d}f. $$
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The associated Poisson bracket on $S$ is then given by:
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$$ \{f,g\}(v):=X_f(g)=-X_g(f).$$
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=== Heisenberg Poisson bracket ===
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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:
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$$ \{f,g\}_M(x,t)=\{f(\cdot,t),g(\cdot,t)\}_S(x).\]
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This is called the '''Heisenberg Poisson bracket'''. Actually the same construction can be performed replacing $S$ by any Poisson manifold.
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=== Linear Poisson brackets ===
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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:
 +
$$ \{f,g\}(v):=\langle [\textrm{d}_v f, \textrm{d}_v], v\rangle. $$
  
 
== Hamiltonian Systems ==
 
== Hamiltonian Systems ==

Revision as of 11:44, 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 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: $$ X_h(f):=\{h,f\}.$$

How to Cite This Entry:
Poisson manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_manifold&oldid=19525