General Poisson manifolds from bivector fields

A real Poisson algebra is a commutative unital algebra over $\mathbb{R}$ equipped with an $\mathbb{R}$-bilinear bracket $\{,\}$ that is skew-symmetric, satisfies the Jacobi identity, and is a derivation in each entry. The algebra-over-a-commutative-ring convention is the one in Algebras over a commutative ring, central structure maps, and algebra homomorphisms.

Let $M$ be a smooth $n$-manifold (Smooth manifolds and their smooth charts). A smooth bivector datum on $M$ assigns to each smooth coordinate chart $x=(x^1,\dots ,x^n)$ smooth functions ${\pi }_x^{ij}$ on its coordinate domain, with ${\pi }_x^{ij}=-{\pi }_x^{ji}$. On an overlap with coordinates $y=(y^1,\dots ,y^n)$, require the contravariant tensor transformation law ${\pi }_y^{ab}(y(x))={\sum }_{i,j=1}^n\frac{\mathrm{\partial }{y}^a}{\mathrm{\partial }{x}^i}(x)\frac{\mathrm{\partial }{y}^b}{\mathrm{\partial }{x}^j}(x){\pi }_x^{ij}(x)\mathrm{.}$ For smooth functions $f,g$ on a coordinate domain, the associated local bracket is $\{f,g{\}}_{\pi ,x}:={\sum }_{i,j=1}^n{\pi }_x^{ij}\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}^i}\frac{\mathrm{\partial }g}{\mathrm{\partial }{x}^j}\mathrm{.}$ A bivector datum is Poisson when this local bracket satisfies the Jacobi identity on every coordinate domain. A Poisson manifold is a smooth manifold with a Poisson bivector datum. The next proposition proves that the local brackets agree on overlaps, are biderivations, and satisfy Jacobi exactly when the Schouten self-bracket vanishes; it justifies this definition as a global Poisson bracket.

Here ${\mathrm{\partial }}_i=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^i$, and repeated coordinate indices in tensor formulas are summed. To fix the Schouten convention needed here, put $J_{\pi }^{ijk}={\pi }^{i\mathrm{\ell }}{\mathrm{\partial }}_{\mathrm{\ell }}{\pi }^{jk}+{\pi }^{j\mathrm{\ell }}{\mathrm{\partial }}_{\mathrm{\ell }}{\pi }^{ki}+{\pi }^{k\mathrm{\ell }}{\mathrm{\partial }}_{\mathrm{\ell }}{\pi }^{ij}\mathrm{.}$ For $\pi =\frac{1}{2}{\pi }^{ij}{\mathrm{\partial }}_i\wedge {\mathrm{\partial }}_j$, the local components of its Schouten–Nijenhuis self-bracket are $[\pi ,\pi {]}_{SN}^{ijk}:=2J_{\pi }^{ijk}$. In this pair, $[\pi ,\pi {]}_{SN}=0$ means that these coefficients vanish in every chart.

No nondegeneracy assumption is imposed on $\pi$. In particular, the zero bivector and bivectors of nonconstant rank are permitted. No global frame of the tangent bundle is chosen.