[[Algebra theory MOC]]
# Poisson algebra

A **Poisson algebra** $(A, \cdot, [\cdot,\cdot])$ is an [[K-monoid]] $(A, \cdot)$ and [[Lie algebra]] $(A, [\cdot,\cdot])$ such that the [[Lie algebra|Lie bracket]] (called the **Poisson bracket**) is a [[Derivation on an algebra|derivation]], #m/def/falg  i.e.
$$
\begin{align*}
[x,yz] = [x,y]z + y[x,z]
\end{align*}
$$
whence follows
$$
\begin{align*}
[xy,z] =  [x,z]y + x[y,z]
\end{align*}
$$


## Examples

- Any [[K-monoid]] with its [[Commutator]] forms a Poisson algebra

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