Lie algebra

Definition
Suppose $$R$$ is a commutative unital ring, i.e., an associative ring whose multiplication is commutative and has an identity element.

A Lie algebra over $$R$$ is a defining ingredient::Lie ring $$L$$ whose additive group is equipped with a $$R$$-module structure and whose Lie bracket is $$R$$-bilinear.

Explicitly, a Lie algebra over $$R$$ is a $$R$$-module $$L$$ equipped with a map $$[ \, \ ]: L \times L \to L$$ satisfying all the following conditions:

Particular cases

 * In the case that $$R = \mathbb{Z}$$, the notion of $$R$$-Lie algebra coincides with the usual notion of Lie ring.

Universal enveloping algebra
Every Lie algebra has a universal enveloping algebra. An enveloping algebra for a Lie algebra is an associative algebra over the same base field which contains the elements of the Lie algebra, such that:


 * The addition in the enveloping algebra is the same as that within the Lie algebra
 * For those elements which are in the Lie algebra, the commutator coincides with the Lie bracket

The universal enveloping algebra is an algebra that is universal among all enveloping algebras.