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: |
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: |
The two versions are equivalent by skew symmetry.
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.