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Basic concepts

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The statement of the theorem

Given a representation ρ:g→End⁡(V)\rho :{\mathfrak {g}}\rightarrow \operatorname {End} (V) of a Lie algebra g{\mathfrak {g}} , we say that a subspace WW of VV is invariant if ρ(X)w∈W\rho (X)w\in W for all w∈Ww\in W and X∈gX\in {\mathfrak {g}} . A nonzero representation is said to be irreducible if the only invariant subspaces are VV itself and the zero space {0}\{0\} . The term simple module is also used for an irreducible representation. Let g{\mathfrak {g}} be a Lie algebra. Let V, W be g{\mathfrak {g}} -modules. Then a linear map f:V→Wf:V\to W is a homomorphism of g{\mathfrak {g}} -modules if it is g{\mathfrak {g}} -equivariant; i.e., f(X⋅v)=X⋅f(v)f(X\cdot v)=X\cdot f(v) for any X∈g, v∈VX\in {\mathfrak {g}},\,v\in V . If f is bijective, V,WV,W 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. A simple but useful tool in studying irreducible representations is Schur's lemma. It has two parts: - If V, W are irreducible g{\mathfrak {g}} -modules and f:V→Wf:V\to W is a homomorphism, then ff is either zero or an isomorphism. - If V is an irreducible g{\mathfrak {g}} -module over an algebraically closed field and f:V→Vf:V\to V is a homomorphism, then ff is a scalar multiple of the identity. Let V be a representation of a Lie algebra g{\mathfrak {g}} . 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 g{\mathfrak {g}} 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. 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 if the adjoint representation is semisimple. Certainly, every (finite-dimensional) semisimple Lie algebra g{\mathfrak {g}} is reductive, since every representation of g{\mathfrak {g}} 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. An element v of V is said to be g{\mathfrak {g}} -invariant if x⋅v=0x\cdot v=0 for all x∈gx\in {\mathfrak {g}} . The set of all invariant elements is denoted by VgV^{\mathfrak {g}} . - ^Hall 2015 Theorem 4.29 - ^Dixmier 1977, Theorem 1.6.3