Editing Lie algebra representation (section)

== Basic concepts ==
===Invariant subspaces and irreducibility===
Given a representation <math>\rho:\mathfrak{g}\rightarrow\operatorname{End}(V)</math> of a Lie algebra <math>\mathfrak{g}</math>, we say that a subspace <math>W</math> of <math>V</math> is '''invariant''' if <math>\rho(X)w\in W</math> for all <math>w\in W</math> and <math>X\in\mathfrak{g}</math>. A nonzero representation is said to be '''irreducible''' if the only invariant subspaces are <math>V</math> itself and the zero space <math>\{0\}</math>. The term ''simple module'' is also used for an irreducible representation.

===Homomorphisms===
Let <math>\mathfrak{g}</math> be a [[Lie algebra]]. Let ''V'', ''W'' be <math>\mathfrak{g}</math>-modules. Then a linear map <math>f: V \to W</math> is a '''homomorphism''' of <math>\mathfrak{g}</math>-modules if it is <math>\mathfrak{g}</math>-equivariant; i.e., <math>f(X\cdot v) = X\cdot f(v)</math> for any <math>X \in \mathfrak{g},\, v \in V</math>. If ''f'' is bijective, <math>V, W</math> are said to be '''equivalent'''. Such maps are also referred to as '''intertwining maps''' or '''morphisms'''.

Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.

===Schur's lemma===
{{main|Schur's lemma}}
A simple but useful tool in studying irreducible representations is Schur's lemma. It has two parts:<ref>{{harvnb|Hall|2015}} Theorem 4.29</ref>
*If ''V'', ''W'' are irreducible <math>\mathfrak{g}</math>-modules and <math>f: V \to W</math> is a homomorphism, then <math>f</math> is either zero or an isomorphism.
*If ''V'' is an irreducible <math>\mathfrak{g}</math>-module over an algebraically closed field and <math>f: V \to V</math> is a homomorphism, then <math>f</math> is a scalar multiple of the identity.

===Complete reducibility===

Let ''V'' be a representation of a Lie algebra <math>\mathfrak{g}</math>. Then ''V'' is said to be '''completely reducible''' (or semisimple) if it is isomorphic to a direct sum of irreducible representations (cf. [[semisimple module]]). If ''V'' is finite-dimensional, then ''V'' is completely reducible if and only if every invariant subspace of ''V'' has an invariant complement. (That is, if ''W'' is an invariant subspace, then there is another invariant subspace ''P'' such that ''V'' is the direct sum of ''W'' and ''P''.)

If <math>\mathfrak{g}</math> is a finite-dimensional [[semisimple Lie algebra]] over a field of characteristic zero and ''V'' is finite-dimensional, then ''V'' is semisimple; this is [[Weyl's complete reducibility theorem]].<ref>{{harvnb|Dixmier|1977|loc=Theorem 1.6.3}}</ref> Thus, for semisimple Lie algebras, a classification of irreducible (i.e. simple) representations leads immediately to classification of all representations. For other Lie algebra, which do not have this special property, classifying the irreducible representations may not help much in classifying general representations.

A Lie algebra is said to be [[Reductive Lie algebra|reductive]] if the adjoint representation is semisimple. Certainly, every (finite-dimensional) semisimple Lie algebra <math>\mathfrak g</math> is reductive, since ''every'' representation of <math>\mathfrak g</math> is completely reducible, as we have just noted. In the other direction, the definition of a reductive Lie algebra means that it decomposes as a direct sum of ideals (i.e., invariant subspaces for the adjoint representation) that have no nontrivial sub-ideals. Some of these ideals will be one-dimensional and the rest are simple Lie algebras. Thus, a reductive Lie algebra is a direct sum of a commutative algebra and a semisimple algebra.

===Invariants===

An element ''v'' of ''V'' is said to be <math>\mathfrak{g}</math>-invariant if <math>x\cdot v = 0</math> for all <math>x \in \mathfrak{g}</math>. The set of all invariant elements is denoted by <math>V^\mathfrak{g}</math>.