Editing Lie algebra (section)

==Definition of a Lie algebra==
A Lie algebra is a vector space <math>\,\mathfrak{g}</math> over a [[field (mathematics)|field]] <math>F</math> together with a [[binary operation]] <math>[\,\cdot\,,\cdot\,]: \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}</math> called the Lie bracket, satisfying the following axioms:{{efn|More generally, one has the notion of a Lie algebra over any [[commutative ring]] ''R'': an ''R''-module with an alternating ''R''-bilinear map that satisfies the Jacobi identity ({{harvtxt|Bourbaki|1989|loc=Section 2}}).}}

* ''Bilinearity'',
::<math> [a x + b y, z] = a [x, z] + b [y, z], </math>
::<math> [z, a x + b y] = a[z, x] + b [z, y] </math>
:for all scalars <math>a,b</math> in <math>F</math> and all elements <math>x,y,z</math> in <math>\mathfrak{g}</math>.

* The ''Alternating'' property,
::<math> [x,x]=0\ </math>
:for all <math>x</math> in <math>\mathfrak{g}</math>.

* The ''Jacobi identity'',
:: <math> [x,[y,z]] + [z,[x,y]] + [y,[z,x]]  = 0 \ </math>
:for all <math>x,y,z</math> in <math>\mathfrak{g}</math>.

Given a Lie group, the Jacobi identity for its Lie algebra follows from the associativity of the group operation.

Using bilinearity to expand the Lie bracket <math> [x+y,x+y] </math> and using the alternating property shows that <math> [x,y] + [y,x]=0 </math> for all <math>x,y</math> in <math>\mathfrak{g}</math>. Thus bilinearity and the alternating property together imply
* [[Anticommutativity]],
:: <math> [x,y] = -[y,x],\ </math>
:for all <math>x,y</math> in <math>\mathfrak{g}</math>. If the field does not have [[Characteristic (algebra)|characteristic]] 2, then anticommutativity implies the alternating property, since it implies <math>[x,x]=-[x,x].</math><ref>{{harvnb|Humphreys|1978|p=1.}}</ref>

It is customary to denote a Lie algebra by a lower-case [[fraktur]] letter such as <math>\mathfrak{g, h, b, n}</math>. If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group's name: for example, the Lie algebra of [[special unitary group|SU(''n'')]] is <math>\mathfrak{su}(n)</math>.

===Generators and dimension===
The ''dimension'' of a Lie algebra over a field means its [[dimension (vector space)|dimension as a vector space]]. In physics, a vector space [[basis (linear algebra)|basis]] of the Lie algebra of a Lie group ''G'' may be called a set of ''generators'' for ''G''. (They are "infinitesimal generators" for ''G'', so to speak.) In mathematics, a set ''S'' of ''generators'' for a Lie algebra <math>\mathfrak{g}</math> means a subset of <math>\mathfrak{g}</math> such that any Lie subalgebra (as defined below) that contains ''S'' must be all of <math>\mathfrak{g}</math>. Equivalently, <math>\mathfrak{g}</math> is spanned (as a vector space) by all iterated brackets of elements of ''S''.