Variety of Lie rings
Definition
As a plain variety
The variety of Lie rings is the variety of algebras with the operator domain consisting of:
- A binary operation
- A unary operation (prefix symbol)
- A constant
- A binary operation
![{\displaystyle [\ ,\ ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7717dae012f042ad66888075df87a1ebac4392a)
such that the following universal identities are satisfied:
![{\displaystyle [x,y+z]=[x,y]+[x,z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47901d7938a7d54bcd888ab02f85174176624563)
![{\displaystyle [x+y,z]=[x,z]+[y,z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8aa9f85abfec5ca88ef367e4529ab117853e16)
![{\displaystyle [x,x]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94ccb967db9ab0773c5aa2e0fc234cd33373f193)
![{\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23655a62f2a7cc545f121d9bcc30fe2c56731457)
The identities (1)-(4) say that we get an Abelian group under , the identities (5) and (6) say the Lie bracket is bilinear, the identity (7) says it is alternating, and the identity (8) is the Jacobi identity.
Properties
A complete listing of the universal algebra-theoretic properties of Lie rings is available at:
Category:Property satisfactions for the variety of Lie rings