Editing Lie algebra (section)

== Representations ==
{{main|Lie algebra representation}}

===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''.

===Adjoint representation===
For any Lie algebra <math>\mathfrak{g}</math>, the [[adjoint representation of a Lie algebra|adjoint representation]] is the representation
:<math>\operatorname{ad}\colon\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})</math>
given by <math>\operatorname{ad}(x)(y) = [x, y]</math>. (This is a representation of <math>\mathfrak{g}</math> by the Jacobi identity.)

===Goals of representation theory===
One important aspect of the study of Lie algebras (especially semisimple Lie algebras, as defined below) is the study of their representations. Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra <math>\mathfrak{g}</math>. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather, the goal is to understand all possible representations of <math>\mathfrak{g}</math>. For a semisimple Lie algebra over a field of characteristic zero, [[Weyl's theorem on complete reducibility|Weyl's theorem]]<ref name="reducibility">{{harvnb|Hall|2015|loc=Theorem 10.9.}}</ref> says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The finite-dimensional irreducible representations are well understood from several points of view; see the [[representation theory of semisimple Lie algebras]] and the [[Weyl character formula]].

===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.

===Representation theory in physics===
The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically, they are the relations of the Lie algebra of the relevant symmetry group. An example is the [[angular momentum operator]]s, whose commutation relations are those of the Lie algebra <math>\mathfrak{so}(3)</math> of the rotation group <math>\mathrm{SO}(3)</math>. Typically, the space of states is far from being irreducible under the pertinent operators, but
one can attempt to decompose it into irreducible pieces. In doing so, one needs to know the irreducible representations of the given Lie algebra. In the study of the [[Hydrogen-like atom|hydrogen atom]], for example, quantum mechanics textbooks classify (more or less explicitly) the finite-dimensional irreducible representations of the Lie algebra <math>\mathfrak{so}(3)</math>.<ref name="quantum" />