Editing Poisson algebra (section)

===Lie algebras===
The [[tensor algebra]] of a [[Lie algebra]] has a Poisson algebra structure. A very explicit construction of this is given in the article on [[universal enveloping algebra]]s.

The construction proceeds by first building the [[tensor algebra]] of the underlying vector space of the Lie algebra. The tensor algebra is simply the [[disjoint union]] ([[direct sum]] ⊕) of all tensor products of this vector space. One can then show that the Lie bracket can be consistently lifted to the entire tensor algebra: it obeys both the product rule, and the Jacobi identity of the Poisson bracket, and thus is the Poisson bracket, when lifted. The pair of products {,} and ⊗ then form a Poisson algebra. Observe that ⊗ is neither commutative nor is it anti-commutative: it is merely associative.  

Thus, one has the general statement that the tensor algebra of any Lie algebra is a Poisson algebra.  The universal enveloping algebra is obtained by modding out the Poisson algebra structure.