Weyl's theorem on complete reducibility

In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let formula_1 be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over formula_1 is semisimple as a module (i.e., a direct sum of simple modules.)
The enveloping algebra is semisimple.
Weyl's theorem implies (in fact is equivalent to) that the enveloping algebra of a finite-dimensional representation is a semisimple ring in the following way.
Given a finite-dimensional Lie algebra representation formula_3, let formula_4 be the associative subalgebra of the endomorphism algebra of "V" generated by formula_5. The ring "A" is called the enveloping algebra of formula_6. If formula_6 is semisimple, then "A" is semisimple. (Proof: Since "A" is a finite-dimensional algebra, it is an Artinian ring; in particular, the Jacobson radical "J" is nilpotent. If "V" is simple, then formula_8 implies that formula_9. In general, "J" kills each simple submodule of "V"; in particular, "J" kills "V" and so "J" is zero.) Conversely, if "A" is semisimple, then "V" is a semisimple "A"-module; i.e., semisimple as a formula_10-module. (Note that a module over a semisimple ring is semisimple since a module is a quotient of a free module and "semisimple" is preserved under the free and quotient constructions.)
Application: preservation of Jordan decomposition.
Here is a typical application.
"Proof": First we prove the special case of (i) and (ii) when formula_6 is the inclusion; i.e., formula_10 is a subalgebra of formula_13. Let formula_14 be the Jordan decomposition of the endomorphism formula_15, where formula_16 are semisimple and nilpotent endomorphisms in formula_17. Now, formula_18 also has the Jordan decomposition, which can be shown (see Jordan–Chevalley decomposition#Lie algebras) to respect the above Jordan decomposition; i.e., formula_19 are the semisimple and nilpotent parts of formula_18. Since formula_19 are polynomials in formula_18 then, we see formula_23. Thus, they are derivations of formula_1. Since formula_1 is semisimple, we can find elements formula_26 in formula_1 such that formula_28 and similarly for formula_29. Now, let "A" be the enveloping algebra of formula_1; i.e., the subalgebra of the endomorphism algebra of "V" generated by formula_10. As noted above, "A" has zero Jacobson radical. Since formula_32, we see that formula_33 is a nilpotent element in the center of "A". But, in general, a central nilpotent belongs to the Jacobson radical; hence, formula_34 and thus also formula_35. This proves the special case.
In general, formula_36 is semisimple (resp. nilpotent) when formula_37 is semisimple (resp. nilpotent). This immediately gives (i) and (ii). formula_38
Proofs.
Analytic proof.
Weyl's original proof (for complex semisimple Lie algebras) was analytic in nature: it famously used the unitarian trick. Specifically, one can show that every complex semisimple Lie algebra formula_1 is the complexification of the Lie algebra of a simply connected compact Lie group formula_40. (If, for example, formula_41, then formula_42.) Given a representation formula_6 of formula_1 on a vector space formula_45 one can first restrict formula_6 to the Lie algebra formula_47 of formula_40. Then, since formula_40 is simply connected, there is an associated representation formula_50 of formula_40. Integration over formula_40 produces an inner product on formula_53 for which formula_50 is unitary. Complete reducibility of formula_50 is then immediate and elementary arguments show that the original representation formula_6 of formula_1 is also completely reducible.
Algebraic proof 1.
Let formula_58 be a finite-dimensional representation of a Lie algebra formula_10 over a field of characteristic zero. The theorem is an easy consequence of Whitehead's lemma, which says formula_60 is surjective, where a linear map formula_61 is a derivation if formula_62. The proof is essentially due to Whitehead.
Let formula_63 be a subrepresentation. Consider the vector subspace formula_64 that consists of all linear maps formula_65 such that formula_66 and formula_67. It has a structure of a formula_1-module given by: for formula_69,
Now, pick some projection formula_71 onto "W" and consider formula_72 given by formula_73. Since formula_74 is a derivation, by Whitehead's lemma, we can write formula_75 for some formula_76. We then have formula_77; that is to say formula_78 is formula_1-linear. Also, as "t" kills formula_80, formula_78 is an idempotent such that formula_82. The kernel of formula_78 is then a complementary representation to formula_80. formula_38
See also Weibel's homological algebra book.
Algebraic proof 2.
Whitehead's lemma is typically proved by means of the quadratic Casimir element of the universal enveloping algebra, and there is also a proof of the theorem that uses the Casimir element directly instead of Whitehead's lemma.
Since the quadratic Casimir element formula_86 is in the center of the universal enveloping algebra, Schur's lemma tells us that formula_86 acts as multiple formula_88 of the identity in the irreducible representation of formula_1 with highest weight formula_90. A key point is to establish that formula_88 is "nonzero" whenever the representation is nontrivial. This can be done by a general argument or by the explicit formula for formula_88.
Consider a very special case of the theorem on complete reducibility: the case where a representation formula_53 contains a nontrivial, irreducible, invariant subspace formula_80 of codimension one. Let formula_95 denote the action of formula_86 on formula_53. Since formula_53 is not irreducible, formula_95 is not necessarily a multiple of the identity, but it is a self-intertwining operator for formula_53. Then the restriction of formula_95 to formula_80 is a nonzero multiple of the identity. But since the quotient formula_103 is a one dimensional—and therefore trivial—representation of formula_1, the action of formula_86 on the quotient is trivial. It then easily follows that formula_95 must have a nonzero kernel—and the kernel is an invariant subspace, since formula_95 is a self-intertwiner. The kernel is then a one-dimensional invariant subspace, whose intersection with formula_80 is zero. Thus, formula_109 is an invariant complement to formula_80, so that formula_53 decomposes as a direct sum of irreducible subspaces:
Although this establishes only a very special case of the desired result, this step is actually the critical one in the general argument.
Algebraic proof 3.
The theorem can be deduced from the theory of Verma modules, which characterizes a simple module as a quotient of a Verma module by a maximal submodule. This approach has an advantage that it can be used to weaken the finite-dimensionality assumptions (on algebra and representation).
Let formula_53 be a finite-dimensional representation of a finite-dimensional semisimple Lie algebra formula_10 over an algebraically closed field of characteristic zero. Let formula_115 be the Borel subalgebra determined by a choice of a Cartan subalgebra and positive roots. Let formula_116. Then formula_117 is an formula_118-module and thus has the formula_118-weight space decomposition:
where formula_121. For each formula_122, pick formula_123 and formula_124 the formula_10-submodule generated by formula_126 and formula_127 the formula_10-submodule generated by formula_117. We claim: formula_130. Suppose formula_131. By Lie's theorem, there exists a formula_132-weight vector in formula_133; thus, we can find an formula_134-weight vector formula_135 such that formula_136 for some formula_137 among the Chevalley generators. Now, formula_138 has weight formula_139. Since formula_140 is partially ordered, there is a formula_122 such that formula_142; i.e., formula_143. But this is a contradiction since formula_144 are both primitive weights (it is known that the primitive weights are incomparable.). Similarly, each formula_145 is simple as a formula_10-module. Indeed, if it is not simple, then, for some formula_147, formula_148 contains some nonzero vector that is not a highest-weight vector; again a contradiction. formula_38