Poisson manifold

Poisson manifolds are a mathematical setup for classical mechanics with finitely many degrees of freedom.

A **Poisson algebra** is a commutative unital associative algebra $A$, in this case over the field of real or complex numbers, equipped with a Lie bracket $\{,\}:A\otimes A\to A$ such that, for any $f\in A$, $\{ f,-\}:A\to A$ is a derivation of $A$ as an associative algebra.

A **Poisson manifold** is a real smooth manifold $M$ equipped with a **Poisson structure**. A Poisson structure is a Lie algebra bracket $\{,\}:C^\infty(M)\times C^\infty(M)\to C^\infty(M)$ on the vector space of smooth functions on $M$ which together with the pointwise multiplication of functions makes it a Poisson algebra. As derivations of $C^\infty(M)$ correspond to smooth tangent vector fields, for each $f\in C^\infty(M)$ there is a vector $X_f$ given by $X_f(g)=\{f,g\}$ and called the **Hamiltonian vector field** corresponding to the function $f$, which is viewed as a classical hamiltonian function.

Alternatively a Poisson structure on a manifold is given by a choice of smooth antisymmetric bivector called a **Poisson bivector** $P\in\Lambda^2 T M$; then $\{f,g\}:=\langle df\otimes dg, P\rangle$.

This induces and is equivalently encoded by the structure of a Poisson Lie algebroid.

A morphism $h:M\to N$ of Poisson manifolds is a morphism of smooth manifolds such that, for all $f,g\in C^\infty(N)$, $\{f\circ h, g\circ h\}_M = \{f,g\}_N$.

Every manifold admits the *trivial Poisson structure* for which the Poisson bracket simply vanishes on all elements.

Every symplectic manifold carries a natural Poisson structure; however, such Poisson manifolds are very special. It is a basic theorem that Poisson structures on a manifold are equivalent to the smooth foliations of the underlying manifold such that each leaf is a symplectic manifold.

The dual to a finite-dimensional Lie algebra has a natural structure of a Poisson manifold, the *Lie-Poisson structure*. Its leaves are called coadjoint orbits.

Given a symplectic manifold $(X,\omega)$ and given a Hamiltonian function $H \colon X \longrightarrow \mathbb{R}$, there is a Poisson bracket on the functions on the smooth path space $[I,X]$ (the “space of histories” or “space of trajectories”), for $I = [0,1]$ the closed interval, which is such that its symplectic leaves are each a copy of $X$, but regarded as the space of initial conditions for evolution with respect to $H$ with a source term added. For more on this see at *off-shell Poisson bracket*.

Every local action functional which admits a Green's function for its equations of motion defines the Peierls bracket on covariant phase space (where in fact it is symplectic) and also “off-shell” on all of configuration space, where it is a genuine Poisson bracket, the canonocal Poisson bracket of the corresponding prequantum field theory.

Kontsevich formality implies that every Poisson manifold has a family of deformation quantizations, parameterized by the Grothendieck-Teichmüller group.

**duality between algebra and geometry in physics**:

**Examples of sequences of local structures**

geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|

$\leftarrow$ differentiation | integration $\to$ | |||||||

smooth functions | derivative | Taylor series | germ | smooth function | ||||

curve (path) | tangent vector | jet | germ of curve | curve | ||||

smooth space | infinitesimal neighbourhood | formal neighbourhood | open neighbourhood | |||||

function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||

arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||

Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||

symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |

Revised on September 15, 2013 18:13:13
by Urs Schreiber
(89.204.137.43)