Lie ring
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In mathematics a Lie ring is a structure related to Lie algebras that can arise as a generalisation of Lie algebras, or through the study of the Lower central series of groups.
[edit] Formal definition
A Lie ring is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity. More specifically we can define a Lie ring L to be an Abelian group with an operation ![[\cdot,\cdot]](../../../math/3/5/8/3587c5df5edf1176ed7afc1f20f5d8a9.png)
that has the following properties:
- Bilinearity:
![[x + y, z] = [x, z] + [y, z], \quad [z, x + y] = [z, x] + [z, y]](../../../math/6/1/2/612225aa111eef27437e38b56c4eb4b3.png)
- for all x, y, z ∈ L.
- The Jacobi identity:
![[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 \quad](../../../math/6/9/e/69ee53d21fe7eb5e7b8ed7938cb8cab0.png)
- for all x, y, z in L.
- For all x in L.
![[x,x]=0 \quad](../../../math/4/b/3/4b3d4565edeb5d1f29e91ea7b7e4655b.png)
[edit] Examples
- Any Lie algebra over a general ring instead of a field is an example of a Lie ring.
- Any associative ring can be made into a Lie ring by defining a bracket operator [x,y] = xy − yx.
- For an example of a Lie ring arising from the study of groups, let G be a group, and let
be a central series in G - that is ![[G_i,G_j] \subseteq G_{i+j}](../../../math/b/0/4/b0430429deb5fd90754c59648ad33dfe.png)
for any i,j. Then
- is a Lie ring with addition supplied by the group operation (which will be commutative in each homogeneous part), and the bracket operation given by
-
- [xGi,yGj] = [x,y]Gi + j
- extended linearly.
