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.

Lie rings need not be Lie groups under addition. Any Lie algebra is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator $[x,y] = xy - yx$ . Conversely to any Lie algebra there is a corresponding ring, called the universal enveloping algebra.

Lie rings are used in the study of finite p-groups through the Lazard correspondence. The lower central factors of a p-group are finite abelian p-groups, so modules over Z/pZ. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives. The Lie ring structure is enriched with another module homomorphism, then pth power map, making the associated Lie ring a so-called restricted Lie ring.

Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and the reducing modulo p to get a Lie algebra over a finite field.

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.