Cremona transformation

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 and linear Poisson structures.

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: $$ \{f,g\}(v):=X_f(g)=-X_g(f).$$

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 the dual 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: $$ \{f,g\}(v):=\langle [\mathrm{d}_v f, \mathrm{d}_v g], v\rangle. $$

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.

Hamiltonian Systems and Symmetries

Hamiltonian vector fields

On a Poisson manifold $(M,\{~,~\})$, any smooth function $h\in C^\infty(M)$ determines a hamiltonian vector field $X_h$ by setting: $$ X_h(f):=\{h,f\}.$$

One calls the function $h$ the hamiltonian. Note that for a symplectic manifold, viewed as a Poisson manifold, this definition is consistent with the old definition. The flow $\Phi^t_{X_h}$ of a hamiltonian vector field preserves the hamiltonian: $$ h\circ \Phi^t_{X_h}=h. $$

On a Poisson manifold $(M,\{~,~\})$, the functions $f\in C^\infty(M)$ for which the hamiltonian vector field $X_f$ vanishes identically are called Casimirs. They form the center of the Lie algebra $(C^\infty(M),\{~,~\})$.

Poisson vector fields

A vector field $X$ on Poisson manifold $(M,\{~,~\})$ is called a Poisson vector field if it is a derivation of the Poisson bracket: \[ X(\{f,g\})=\{X(f),g\}+\{f,X(g)\}.\] The Jacobi identity shows that any hamiltonian vector field is a Poisson vector field. If $\Phi^t_X$ denotes the flow of the vector field $X$, then $X$ is a Poisson vector field if and only if $\Phi^t_X$ is a 1-paremeter group of Poisson diffeomorphisms.

The vector space $H^1_\pi(M)$ formed by the quotient of the Poisson vector fields modulo hamiltonian vector fields is called the first Poisson cohomology of $M$.

Moment maps

Let $G$ be a Lie group which acts smoothly on a Poisson manifold $(M,\{~,~\})$. We say that $G$ is a symmetry group or that $G\times M \to M$ is a Poisson action iif the action is by Poisson diffeomorphisms. If $G$ is connected and $\rho:\mathfrak{g}\to \mathcal{X}(M)$ is the corresponding infinitesimal action, then the group is a symmetry group if and on if each vector field $\rho(\xi)$ is a Poisson vector field.

A hamiltonian action $G\times M \to M$ is a Poisson action such that the vector fields $\rho(\xi)$ are hamiltonian vector fields: $$ \rho(\xi)=X_{\mu^*(\xi)}, $$ for some smooth $G$-equivariant map $\mu:M\to \mathfrak{g}^*$. Here $\mu^*:\mathfrak{g}\to C^\infty(M)$ denotes the map $\mu^*(\xi)(x)=\langle \mu(x),\xi\rangle$. One calls $\mu$ the moment map.

Constructions with Poisson manifolds

There are many constructions which produce new Poisson manifolds out of old ones.

Poisson submanifolds

Let $(M,\{~,~\})$ be a Poisson manifold and suppose $N\subset M$ is a submanifold with the property that for any $f\in C^\infty(M)$ the hamiltonian vector field $X_f$ is tangent to $N$. Then we have an induced Poisson bracket on $N$ defined by: $$ \{f,g\}_N=\{F,G\}|_N, \forall f,g\in C^\infty(N),$$ where $F,G\in C^\infty(M)$ are any extensions of $f$ and $g$ to $M$: $F|_N=f$ and $G|_N=g$.

Product of Poisson manifolds

If $(M,\{~,~\}_M)$ and $(N,\{~,~\}_N)$ are two Poisson manifolds then their product is the Poisson manifold $(M\times N,\{~,~\}_{M\times N})$ where the Poisson bracket is defined by: $$ \{f,g\}_{M\times N}(x,y):=\{f(\cdot,y),g(\cdot,y)\}_M(x)+\{f(x,\cdot),g(x,\cdot)\}_N(y), \qquad \forall (x,y)\in M\times N.$$ This is the unique Poisson bracket for which the projections $\pi_M:M\times N\to M$ and $\pi_N:M\times N\to N$ are Poisson maps.

Poisson quotients

If $(M,\{~,~\}_M)$ is a Poisson manifold and $G\times M\to M$ is a smooth Lie group action by Poisson diffeomorphisms then the Poisson bracket of any two $G$-invariant functions $f,g\in C^\infty(M)^G$ is again a $G$-invariant function: $\{f,g\}\in C^\infty(M)^G$.

When the action is free and proper, $M/G$ is a smooth manifold and $C^\infty(M/G)\equiv C^\infty(M)^G$, so it follows that $M/G$ carries a natural Poisson bracket $\{~,~\}_{M/G}$. It is the unique Poisson bracket for which the quotient map $q:M\to M/G$ is a Poisson map.

References

• A. Cannas da Silva, A. Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, 10. American Mathematical Society, Providence, RI, 1999. ISBN: 0-8218-0952-0
• J.P. Dufour, N.T. Zung, Poisson structures and their normal forms, Progress in Mathematics, 242. Birkhäuser Verlag, Basel, 2005. ISBN: 978-3-7643-7334-4
• A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geom. 12 (1977), n. 2, 253–300.
• A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983), n.3, 523–557 (Errata and addenda J. Diff. Geom. 22 (1985), 255.)