Let $(X,\mathcal{D})$ be a diffeological space, and let $\alpha\in\Omega^{k}(X)$ and $\beta\in\Omega^{l}(X)$ . Then define the wedge product of $\alpha$ and $\beta$ , denoted $\alpha\wedge\beta$ , to be the $(k+l)$ -form defined by

$$(\alpha\wedge\beta)(p)=\alpha(p)\wedge\beta(p)$$

for all plots $p\in\mathcal{D}$ . Then $\Omega^{*}(X)=\bigoplus_{k=0}^{\infty}\Omega^{k}(X)$ is an exterior algebra.

Let $(X,\mathcal{D})$ be a diffeological space, and let $\alpha$ be a $k$ -form on it. Define the exterior derivative of $\alpha$ , denoted $d\alpha$ , by

$$p^{*}(d\alpha)=d(p^{*}\alpha)$$

for any plot $p\in\mathcal{D}$ . The exterior derivative commutes with pullback, and all of the usual formulae involving pullbacks, the exterior derivative, and the wedge product hold. We thus have the de Rham complex $(\Omega^{*}(X),d)$ .

Given a point $x\in S$ , a derivation of ${C^{\infty}}(S)$ at $x$ is a linear map $v:{C^{\infty}}(S)\to{\mathbb{R}}$ that satisfies Leibniz’ rule: for all $f,g\in{C^{\infty}}(S)$ ,

$$v(fg)=f(x)v(g)+g(x)v(f).$$

The set of all derivations of ${C^{\infty}}(S)$ at $x$ forms a vector space, called the (Zariski) tangent space of $x$ , and is denoted $T_{x}S$ . Define the (Zariski) tangent bundle $TS$ to be the (disjoint) union

$$TS:=\bigcup_{x\in S}T_{x}S.$$

Denote the canonical projection $TS\to S$ by $\tau$ .

A Poisson bracket on a differential structure $\mathcal{F}$ on a differential space $X$ is a Lie bracket ${\{,\}}$ satisfying for any $f,g,h\in\mathcal{F}$ :

$${\{f,gh\}}=h{\{f,g\}}+g{\{f,h\}}.$$