The sea of Poisson Structures and its Graph Complex flows

Index » Poisson structures (intro)

Hamilton's equations

Hamilton's equations in canonical coordinates for a time-independent hamiltonian $H: \mathbb{R}^2 \to \mathbb{R}$ read as follows:

$\left\{\begin{aligned}\dot{q} &= +\frac{\partial H}{\partial p}\\\dot{p} &= -\frac{\partial H}{\partial q}\end{aligned}\right.$

where $(q,p)$ are coordinates on $\mathbb{R}^2$, and dots denote derivatives with respect to time.

Classical Poisson bracket

From the chain rule for derivatives, it follows that the time evolution of a function $F: \mathbb{R}^2 \to \mathbb{R}$ is:

$\begin{aligned} \dot{F} &= \frac{\partial F}{\partial q}\cdot \dot{q} + \frac{\partial F}{\partial p} \cdot \dot{p} &= \qquad \frac{\partial F}{\partial q} \cdot \frac{\partial H}{\partial p} - \frac{\partial F}{\partial p} \cdot \frac{\partial H}{\partial q} \eqqcolon \{F, H\}, \end{aligned}$

this is called the Poisson bracket $\{F,H\}$ of $F$ and $H$.

The Jacobi identity holds: $\{\{F,G\},H\} + \{\{G,H\},F\} + \{\{H,F\},G\} = 0$.

More general Poisson brackets

More generally, a Poisson bracket on $M = \mathbb{R}^d$ is a bracket $\{-,-\}$ on the algebra of smooth real-valued functions $C^\infty(M)$ such that:

Bi-derivation: $\{f,gh\} = \{f,g\}h + g\{f, h\}$,
(Bi-)linearity: $\{f,ag+bh\} = a\{f,g\} + b\{f,h\}$ for $a,b \in \mathbb{R}$,
Skew-symmetry: $\{f,g\} = -\{g,f\}$,
Jacobi identity: $\{\{f,g\},h\} + \{\{g,h\},f\} + \{\{h,f\},g\} = 0$.

Structure coefficients

The structure coefficients of a Poisson bracket in coordinates $x^1,\ldots,x^d$ are the functions $P^{ij} \coloneqq \{x^i, x^j\} \in C^\infty(M)$. They satisfy:

$P^{ij} = -P^{ji}$ for all $i,j=1,\ldots,d$ (skew-symmetry) and
${\sum}_{s=1}^d (P^{si} \partial_s P^{jk} + P^{sj} \partial_s P^{ki} + P^{sk}\partial_s P^{ij}) = 0$ for all $i,j,k=1,\ldots,d$ (Jacobi identity).

Conversely, a collection of such structure coefficients $(P^{ij})$ defines a Poisson bracket on $C^\infty(M)$ by $\{f,g\} = \sum_{i,j=1}^d P^{ij} \cdot \partial_i(f\,) \cdot \partial_j(g\,)$, where $\partial_i \coloneqq \frac{\partial}{\partial x^i}$.

The tensor $P = \sum_{i,j=1}^d P^{ij} \partial_i \wedge \partial_j$ is a bi-vector field, and the Jacobi identity is equivalent to $[P,P] = 0$ where the bracket is the Schouten bracket (a natural extension of the Lie bracket of vector fields).

Deformations of Poisson structures

Suppose we want to formally deform a Poisson structure $P$ by $P(\varepsilon) = P + \varepsilon Q + O(\varepsilon^2)$ where $\varepsilon$ is a formal parameter and $Q$ is a bi-vector field.

From expanding $[P(\varepsilon), P(\varepsilon)] = O(\varepsilon^2)$ we obtain the condition $[P, Q] = 0$, where the bracket is again the Schouten bracket.

Trivial solutions to this deformation problem are those of the form $Q = [P,X]$ for a vector field $X$.