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 $[\cdot,\cdot]$ that has the following properties:

Bilinearity:

$[x + y, z] = [x, z] + [y, z], \quad [z, x + y] = [z, x] + [z, y]$

for all x, y, z ∈ L.

The Jacobi identity:

$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 \quad$

for all x, y, z in L.

For all x in L.

$[x,x]=0 \quad$

[edit] Examples

Any Lie algebra over a general ring instead of a field is an example of a Lie ring. Lie rings are not Lie groups under addition, despite the name.

Any associative ring can be made into a Lie ring by defining a bracket operator $[x, y] = x y - y x$ .

For an example of a Lie ring arising from the study of groups, let $G$ be a group with $(x, y) = x - 1 y - 1 x y$ the commutator operation, and let $G = G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots \supseteq G_n \supseteq \cdots$ be a central series in $G$ — that is the commutator subgroup $(G i, G j)$ is contained in $G i + j$ for any $i, j$ . Then