# Lie algebra

From Groupprops

## Definition

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

A **Lie algebra** over is a Lie ring whose additive group is equipped with a -module structure and whose Lie bracket is -bilinear.

Explicitly, a **Lie algebra** over is a -module equipped with a map satisfying **all** the following conditions:

Condition name | Explicit identities (all variable letters are universally quantified over and variable is universally quantified over ) |
---|---|

-bilinear | Additive in left coordinate: Additive in right coordinate: -scalars can be pulled out of left coordinate: -scalars can be pulled out of right coordinate: |

alternating (hence skew-symmetric) | Alternation: Skew symmetry: The second condition (skew symmetry) follows from the first (alternation); the reverse implication holds only if 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: Right-normed version: The two versions are equivalent by skew symmetry. |

## Particular cases

- In the case that , the notion of -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.