Editing Lie algebra (section)

===Universal enveloping algebra===
{{main|Universal enveloping algebra}}
The functor that takes an associative algebra ''A'' over a field ''F'' to ''A'' as a Lie algebra (by <math>[X,Y]:=XY-YX</math>) has a [[left adjoint]] <math>\mathfrak{g}\mapsto U(\mathfrak{g})</math>, called the '''universal enveloping algebra'''. To construct this: given a Lie algebra <math>\mathfrak{g}</math> over ''F'', let
:<math>T(\mathfrak{g})=F\oplus \mathfrak{g} \oplus (\mathfrak{g}\otimes\mathfrak{g}) \oplus (\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g})\oplus \cdots</math>
be the [[tensor algebra]] on <math>\mathfrak{g}</math>, also called the free associative algebra on the vector space <math>\mathfrak{g}</math>. Here <math>\otimes</math> denotes the [[tensor product]] of ''F''-vector spaces. Let ''I'' be the two-sided [[ideal (ring theory)|ideal]] in <math>T(\mathfrak{g})</math> generated by the elements <math>XY-YX-[X,Y]</math> for <math>X,Y\in\mathfrak{g}</math>; then the universal enveloping algebra is the quotient ring <math>U(\mathfrak{g}) = T(\mathfrak{g}) / I</math>. It satisfies the [[Poincaré–Birkhoff–Witt theorem]]: if <math>e_1,\ldots,e_n</math> is a basis for <math>\mathfrak{g}</math> as an ''F''-vector space, then a basis for <math>U(\mathfrak{g})</math> is given by all ordered products <math>e_1^{i_1}\cdots e_n^{i_n}</math> with <math>i_1,\ldots,i_n</math> natural numbers. In particular, the map <math>\mathfrak{g}\to U(\mathfrak{g})</math> is [[injective]].<ref>{{harvnb|Humphreys|1978|loc=section 17.3.}}</ref>

Representations of <math>\mathfrak{g}</math> are equivalent to [[module (mathematics)|modules]] over the universal enveloping algebra. The fact that <math>\mathfrak{g}\to U(\mathfrak{g})</math> is injective implies that every Lie algebra (possibly of infinite dimension) has a faithful representation (of infinite dimension), namely its representation on <math>U(\mathfrak{g})</math>. This also shows that every Lie algebra is contained in the Lie algebra associated to some associative algebra.