Lie ring
This article defines a non-associative ring property: a property that an be evaluated to true or false for any non-associative ring.
View other non-associative ring properties
Definition
A Lie ring is a set equipped with the structure of an abelian group (operation denoted ) and a bracket operation (called a Lie bracket) satisfying the following additional conditions:
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| Condition name | Explicit identities (all variable letters are universally quantified over ) |
|---|---|
| -bilinear, also known as biadditive, also known as left and right distributive | Additive in left coordinate: Additive in 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. |
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![{\displaystyle [x,y+z]=[x,y]+[x,z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47901d7938a7d54bcd888ab02f85174176624563)
![{\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)
A Lie ring which is also an algebra over a field (or a commutative unital ring) is termed a Lie algebra over that field (or commutative unital ring).
If is a commutative unital ring and is additionally equipped with the structure of a -module, and the Lie bracket of is -bilinear, then this makes a -Lie algebra.
Viewpoints
- The Lie rings form a variety of algebras called the variety of Lie rings.
- The Jacobi identity for Lie rings is a close counterpart to associativity in groups.
- The associated Lie ring for a strongly central series and the associated Lie ring for a group form a convenient way to pass from groups to Lie rings.
- The Malcev correspondence and its generalization, the Lazard correspondence, form convenient ways to move back and forth between groups and Lie rings without loss of information.
Related notions
- Lie algebra is a variation where the underlying group is a module over a specified commutative unital ring and the Lie bracket is a bilinear map.
- Multiplicative Lie ring is a generalization to cases where the underlying group is possibly non-abelian.