Poisson algebra

A Poisson algebra $(𝐴, \cdot, [\cdot, \cdot])$ is a $K$ -monoid $(𝐴, \cdot)$ and Lie $K$ -algebra $(𝐴, [\cdot, \cdot])$ such that the Lie bracket (called the Poisson bracket) is a derivation, #m/def/ralg i.e.

[𝑥, 𝑦 𝑧] = [𝑥, 𝑦] 𝑧 + 𝑦 [𝑥, 𝑧]

whence follows

[𝑥 𝑦, 𝑧] = [𝑥, 𝑧] 𝑦 + 𝑥 [𝑦, 𝑧]

Examples

Any $K$ -monoid with its Commutator forms a Poisson algebra

#state/tidy | #lang/en | #SemBr