Editing Lie algebra representation (section)

== Enveloping algebras ==
{{main|Universal enveloping algebra}}
To each Lie algebra <math>\mathfrak{g}</math> over a field ''k'', one can associate a certain [[ring (mathematics)|ring]] called the universal enveloping algebra of <math>\mathfrak{g}</math> and denoted <math>U(\mathfrak{g})</math>. The universal property of the universal enveloping algebra guarantees that every representation of <math>\mathfrak{g}</math> gives rise to a representation of <math>U(\mathfrak{g})</math>. Conversely, the [[Poincaré–Birkhoff–Witt theorem|PBW theorem]] tells us that <math>\mathfrak{g}</math> sits inside <math>U(\mathfrak{g})</math>, so that every representation of <math>U(\mathfrak{g})</math> can be restricted to <math>\mathfrak{g}</math>. Thus, there is a one-to-one correspondence between representations of <math>\mathfrak{g}</math> and those of <math>U(\mathfrak{g})</math>.

The universal enveloping algebra plays an important role in the representation theory of semisimple Lie algebras, described above. Specifically, the finite-dimensional irreducible representations are constructed as quotients of [[Verma module]]s, and Verma modules are constructed as quotients of the universal enveloping algebra.<ref>{{harvnb|Hall|2015}} Section 9.5</ref>

The construction of <math>U(\mathfrak{g})</math> is as follows.<ref>{{harvnb|Jacobson|1962}}</ref> Let ''T'' be the [[tensor algebra]] of the vector space <math>\mathfrak{g}</math>. Thus, by definition, <math>T = \oplus_{n=0}^\infty \otimes_1^n \mathfrak{g}</math> and the multiplication on it is given by <math>\otimes</math>. Let <math>U(\mathfrak{g})</math> be the [[quotient ring]] of ''T'' by the ideal generated by elements of the form 
:<math>[X, Y] - (X \otimes Y - Y \otimes X)</math>. 
There is a natural linear map from <math>\mathfrak{g}</math> into <math>U(\mathfrak{g})</math> obtained by restricting the quotient map of <math>T \to U(\mathfrak{g})</math> to degree one piece. The [[PBW theorem]] implies that the canonical map is actually injective. Thus, every Lie algebra <math>\mathfrak{g}</math> can be embedded into an associative algebra <math>A=U(\mathfrak{g})</math>in such a way that the bracket on <math>\mathfrak{g}</math> is given by <math>[X,Y]=XY-YX</math> in <math>A</math>.

If <math>\mathfrak{g}</math> is [[abelian Lie algebra|abelian]], then <math>U(\mathfrak{g})</math> is the symmetric algebra of the vector space <math>\mathfrak{g}</math>.

Since <math>\mathfrak{g}</math> is a module over itself via adjoint representation, the enveloping algebra <math>U(\mathfrak{g})</math> becomes a <math>\mathfrak{g}</math>-module by extending the adjoint representation. But one can also use the left and right [[regular representation]] to make the enveloping algebra a <math>\mathfrak{g}</math>-module; namely, with the notation <math>l_X(Y) = XY, X \in \mathfrak{g}, Y \in U(\mathfrak{g})</math>, the mapping <math>X \mapsto l_X</math> defines a representation of <math>\mathfrak{g}</math> on <math>U(\mathfrak{g})</math>. The right regular representation is defined similarly.