Editing Lie algebra (section)

===Definitions===
Given a vector space ''V'', let <math>\mathfrak{gl}(V)</math> denote the Lie algebra consisting of all linear maps from ''V'' to itself, with bracket given by <math>[X,Y]=XY-YX</math>. A ''representation'' of a Lie algebra <math>\mathfrak{g}</math> on ''V'' is a Lie algebra homomorphism
:<math>\pi\colon \mathfrak g \to \mathfrak{gl}(V).</math>
That is, <math>\pi</math> sends each element of <math>\mathfrak{g}</math> to a linear map from ''V'' to itself, in such a way that the Lie bracket on <math>\mathfrak{g}</math> corresponds to the commutator of linear maps.

A representation is said to be ''faithful'' if its kernel is zero. [[Ado's theorem]] states that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful representation on a finite-dimensional vector space. [[Kenkichi Iwasawa]] extended this result to finite-dimensional Lie algebras over a field of any characteristic.<ref>{{harvnb|Jacobson|1979|loc=Ch. VI.}}</ref> Equivalently, every finite-dimensional Lie algebra over a field ''F'' is isomorphic to a Lie subalgebra of <math>\mathfrak{gl}(n,F)</math> for some positive integer ''n''.