# 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.