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 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:

Condition name	Explicit identities (all variable letters $x,y,z$ are universally quantified over $L$ and variable $r$ is universally quantified over $R$)
$R$-bilinear	Additive in left coordinate: $[x+y,z]=[x,z]+[y,z]$ Additive in right coordinate: $[x,y+z]=[x,y]+[x,z]$ $R$-scalars can be pulled out of left coordinate: $[rx,y]=r[x,y]$ $R$-scalars can be pulled out of right coordinate: $[x,ry]=r[x,y]$
alternating (hence skew-symmetric)	Alternation: $[x,x]=0$ Skew symmetry: $[x,y]+[y,x]=0$ The second condition (skew symmetry) follows from the first (alternation); the reverse implication holds only if $L$ is 2-torsion-free. Note also that skew symmetry means that we need assume only one of the two additivity identities and it implies the other.
Jacobi identity	Left-normed version: $[[x,y],z]+[[y,z],x]+[[z,x],y]=0$ Right-normed version: $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ The two versions are equivalent by skew symmetry.

Particular cases

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

Facts

Universal enveloping algebra

Further information: 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.