 Poisson algebra

A Poisson algebra has two compatible products on one vector space: a commutative, associative multiplication and a Lie bracket $\{f,g\}$. Compatibility is the Leibniz ruleThus fixing $f$ makes $g\mapsto \{f,g\}$ an algebraic derivation. The Lie bracket also satisfies the Jacobi identity; an arbitrary bilinear bracket satisfying antisymmetry of a Lie bracket and the product rule need not do so.^[ref]

For example, the polynomial ring $\mathbb{R}[q,p]$ becomes a Poisson algebra with the canonical Poisson bracketThe partial derivatives give $\{q,p\}=1$ and $\{q,p^2\}=2p$, illustrating how the Leibniz rule extends the bracket from generators to their products. More generally, smooth functions on a symplectic manifold form a Poisson algebra. A finite-dimensional Lie algebra of moment map components need not be closed under ordinary multiplication: adding all their products normally gives a much larger Poisson algebra.