Editing Lie algebra (section)

== Lie ring ==
The definition of a Lie algebra over a field extends to define a Lie algebra over any [[commutative ring]] ''R''. Namely, a Lie algebra <math>\mathfrak{g}</math> over ''R'' is an ''R''-[[module (mathematics)|module]] with an alternating ''R''-bilinear map <math>[\ , \ ]\colon \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}</math> that satisfies the Jacobi identity. A Lie algebra over the ring <math>\mathbb{Z}</math> of [[integer]]s is sometimes called a '''Lie ring'''. (This is not directly related to the notion of a Lie group.)

Lie rings are used in the study of finite [[p-group]]s (for a prime number ''p'') through the ''Lazard correspondence''.<ref>{{harvnb|Khukhro|1998|loc=Ch. 6.}}</ref> The lower central factors of a finite ''p''-group are finite abelian ''p''-groups. 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; see the example below.

[[p-adic analytic group|p-adic Lie groups]] are related to Lie algebras over the field <math>\mathbb{Q}_p</math> of [[p-adic number]]s as well as over the ring <math>\mathbb{Z}_p</math> of [[p-adic integer]]s.<ref>{{harvnb|Serre|2006|loc=Part II, section V.1.}}</ref> Part of [[Claude Chevalley]]'s construction of the finite [[groups of Lie type]] involves showing that a simple Lie algebra over the complex numbers comes from a Lie algebra over the integers, and then (with more care) a [[group scheme]] over the integers.<ref>{{harvnb|Humphreys|1978|loc=section 25.}}</ref>

=== Examples ===
* Here is a construction of Lie rings arising from the study of abstract groups. For elements <math>x,y</math> of a group, define the commutator <math>[x,y]= x^{-1}y^{-1}xy</math>. Let <math>G = G_1 \supseteq G_2 \supseteq G_3 \supseteq \cdots \supseteq G_n \supseteq \cdots</math> be a ''filtration'' of a group <math>G</math>, that is, a chain of subgroups such that <math>[G_i,G_j]</math> is contained in <math>G_{i+j}</math> for all <math>i,j</math>. (For the Lazard correspondence, one takes the filtration to be the lower central series of ''G''.) Then
:: <math>L = \bigoplus_{i\geq 1} G_i/G_{i+1}</math>
:is a Lie ring, with addition given by the group multiplication (which is abelian on each quotient group <math>G_i/G_{i+1}</math>), and with Lie bracket <math>G_i/G_{i+1} \times G_j/G_{j+1} \to G_{i+j}/G_{i+j+1}</math> given by commutators in the group:<ref>{{harvnb|Serre|2006|loc=Part I, Chapter II.}}</ref>
:: <math>[xG_{i+1}, yG_{j+1}] := [x,y]G_{i+j+1}. </math>

:For example, the Lie ring associated to the lower central series on the [[dihedral group]] of order 8 is the Heisenberg Lie algebra of dimension 3 over the field <math>\mathbb{Z}/2\mathbb{Z}</math>.