Lie ring

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.

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.

$[x,x]=0 \quad$

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 $[x,y] = xy - yx$.

• For an example of a Lie ring arising from the study of groups, let $G$ be a group with $(x,y) = x^{-1}y^{-1}xy$ 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

$L = \bigoplus G_i/G_{i+1}$

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

$[xG_i, yG_j] = (x,y)G_{i+j}\$

extended linearly. Note that the centrality of the series ensures the commutator $(x,y)$ gives the bracket operation the appropriate Lie theoretic properties.