Lie algebra
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:
![{\displaystyle [\ ,\ ]:L\times L\to L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e4d327f4b1c6cf7dccc24d69031f94e3609284)
| 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. |
![{\displaystyle [x+y,z]=[x,z]+[y,z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8aa9f85abfec5ca88ef367e4529ab117853e16)
![{\displaystyle [x,y+z]=[x,y]+[x,z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47901d7938a7d54bcd888ab02f85174176624563)
![{\displaystyle [rx,y]=r[x,y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f30e4d321c6bd1f8b7c6952f16883c77c85e2121)
![{\displaystyle [x,ry]=r[x,y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b565db6a99c52504437c56ab47e090806c83ba0)
![{\displaystyle [x,x]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94ccb967db9ab0773c5aa2e0fc234cd33373f193)
![{\displaystyle [x,y]+[y,x]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba3f4019700f26de2bd59625e7b078ba3d11929)
![{\displaystyle [[x,y],z]+[[y,z],x]+[[z,x],y]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95eabacf905ea58f824b3b3da0bfc319b3be2590)
![{\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23655a62f2a7cc545f121d9bcc30fe2c56731457)
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.