Editing Lie algebra (section)

== Basic examples ==
===Abelian Lie algebras===
A Lie algebra is called '''abelian''' if its Lie bracket is identically zero. Any vector space <math>V</math> endowed with the identically zero Lie bracket becomes a Lie algebra. Every one-dimensional Lie algebra is abelian, by the alternating property of the Lie bracket.

=== The Lie algebra of matrices ===
* On an associative algebra <math>A</math> over a field <math>F</math> with multiplication written as <math>xy</math>, a Lie bracket may be defined by the commutator <math>[x,y] = xy - yx</math>. With this bracket, <math>A</math> is a Lie algebra. (The Jacobi identity follows from the associativity of the multiplication on <math>A</math>.) <ref>{{harvnb|Bourbaki|1989|loc=§1.2. Example 1.}}</ref>
* The [[endomorphism ring]] of an <math>F</math>-vector space <math>V</math> with the above Lie bracket is denoted <math>\mathfrak{gl}(V)</math>.
*For a field ''F'' and a positive integer ''n'', the space of ''n'' × ''n'' [[matrix (mathematics)|matrices]] over ''F'', denoted <math>\mathfrak{gl}(n, F)</math> or <math>\mathfrak{gl}_n(F)</math>, is a Lie algebra with bracket given by the commutator of matrices: <math>[X,Y]=XY-YX</math>.<ref>{{harvnb|Bourbaki|1989|loc=§1.2. Example 2.}}</ref> This is a special case of the previous example; it is a key example of a Lie algebra. It is called the '''general linear''' Lie algebra.

:When ''F'' is the real numbers, <math>\mathfrak{gl}(n,\mathbb{R})</math> is the Lie algebra of the [[general linear group]] <math>\mathrm{GL}(n,\mathbb{R})</math>, the group of [[invertible matrix|invertible]] ''n'' x ''n''  real matrices (or equivalently, matrices with nonzero [[determinant]]), where the group operation is matrix multiplication. Likewise, <math>\mathfrak{gl}(n,\mathbb{C})</math> is the Lie algebra of the complex Lie group <math>\mathrm{GL}(n,\mathbb{C})</math>. The Lie bracket on <math>\mathfrak{gl}(n,\R)</math> describes the failure of commutativity for matrix multiplication, or equivalently for the composition of [[linear map]]s. For any field ''F'', <math>\mathfrak{gl}(n,F)</math> can be viewed as the Lie algebra of the [[algebraic group]] <math>\mathrm{GL}(n)</math> over ''F''.