Weyl's theorem on complete reducibility explained

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

ak{g} be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over ak{g} is semisimple as a module (i.e., a direct sum of simple modules.)^[1]

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

\pi: ak{g} \to ak{gl}(V) , let A \subset \operatorname{End}(V) be the associative subalgebra of the endomorphism algebra of V generated by \pi( ak g) . The ring A is called the enveloping algebra of \pi . If \pi 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 JV \subset V implies that JV = 0 . 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 ak g -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

\pi is the inclusion; i.e., ak g is a subalgebra of ak{gl} _{n} = ak{gl}(V) . Let x = S + N be the Jordan decomposition of the endomorphism x , where S, N are semisimple and nilpotent endomorphisms in ak{gl} _{n} . Now, \operatorname{ad} _{ak{gl} _{n}(x)} also has the Jordan decomposition, which can be shown (see Jordan–Chevalley decomposition) to respect the above Jordan decomposition; i.e., \operatorname{ad} _{ak{gl} _{n}(S),} \operatorname{ad} _{ak{gl} _{n}(N)} are the semisimple and nilpotent parts of \operatorname{ad} _{ak{gl} _{n}(x)} . Since \operatorname{ad} _{ak{gl} _{n}(S),} \operatorname{ad} _{ak{gl} _{n}(N)} are polynomials in \operatorname{ad} _{ak{gl} _{n}(x)} then, we see \operatorname{ad} _{ak{gl} _{n}(S),} \operatorname{ad} _{ak{gl} _{n}(N)} : ak g \to ak g . Thus, they are derivations of ak{g} . Since ak{g} is semisimple, we can find elements s, n in ak{g} such that [y, S] = [y, s], y \in ak{g} and similarly for n . Now, let A be the enveloping algebra of ak{g} ; i.e., the subalgebra of the endomorphism algebra of V generated by ak g . As noted above, A has zero Jacobson radical. Since [y, N - n] = 0 , we see that N - n is a nilpotent element in the center of A. But, in general, a central nilpotent belongs to the Jacobson radical; hence, N = n and thus also S = s . This proves the special case.

In general,

\pi(x) is semisimple (resp. nilpotent) when \operatorname{ad}(x) is semisimple (resp. nilpotent). This immediately gives (i) and (ii). \square

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

ak{g} is the complexification of the Lie algebra of a simply connected compact Lie group K .^[2] (If, for example, ak{g}= sl (n; C ) , then K = SU (n) .) Given a representation \pi of ak{g} on a vector space V, one can first restrict \pi to the Lie algebra ak{k} of K . Then, since K is simply connected,^[3] there is an associated representation \Pi of K . Integration over K produces an inner product on V for which \Pi is unitary.^[4] Complete reducibility of \Pi is then immediate and elementary arguments show that the original representation \pi of ak{g} is also completely reducible.

Algebraic proof 1

Let

(\pi, V) be a finite-dimensional representation of a Lie algebra ak g over a field of characteristic zero. The theorem is an easy consequence of Whitehead's lemma, which says V \to \operatorname{Der}( ak g, V), v \mapsto ⋅ v is surjective, where a linear map f: ak g \to V is a derivation if f([x, y]) = x ⋅ f(y) - y ⋅ f(x) . The proof is essentially due to Whitehead.

Let

W \subset V be a subrepresentation. Consider the vector subspace L _{W} \subset \operatorname{End}(V) that consists of all linear maps t: V \to V such that t(V) \subset W and t(W) = 0 . It has a structure of a ak{g} -module given by: for x \in ak{g}, t \in L _{W} ,x ⋅ t = [\pi(x), t] .Now, pick some projection p : V \to V onto W and consider f : ak{g} \to L _{W} given by f(x) = [p, \pi(x)] . Since f is a derivation, by Whitehead's lemma, we can write f(x) = x ⋅ t for some t \in L _{W} . We then have [\pi(x), p + t] = 0 , x \in ak{g} ; that is to say p + t is ak{g} -linear. Also, as t kills W , p + t is an idempotent such that (p + t)(V) = W . The kernel of p + t is then a complementary representation to W . \square

Algebraic proof 2

Whitehead's lemma is typically proved by means of the quadratic Casimir element of the universal enveloping algebra,^[5] 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

C is in the center of the universal enveloping algebra, Schur's lemma tells us that C acts as multiple c _{λ} of the identity in the irreducible representation of ak{g} with highest weight λ . A key point is to establish that c _{λ} is nonzero whenever the representation is nontrivial. This can be done by a general argument ^[6] or by the explicit formula for c _{λ} .

Consider a very special case of the theorem on complete reducibility: the case where a representation

V contains a nontrivial, irreducible, invariant subspace W of codimension one. Let C _{V} denote the action of C on V . Since V is not irreducible, C _{V} is not necessarily a multiple of the identity, but it is a self-intertwining operator for V . Then the restriction of C _{V} to W is a nonzero multiple of the identity. But since the quotient V/W is a one dimensional—and therefore trivial—representation of ak{g} , the action of C on the quotient is trivial. It then easily follows that C _{V} must have a nonzero kernel—and the kernel is an invariant subspace, since C _{V} is a self-intertwiner. The kernel is then a one-dimensional invariant subspace, whose intersection with W is zero. Thus, ker (V _{C)} is an invariant complement to W , so that V decomposes as a direct sum of irreducible subspaces:V = W ⊕ ker (C _{V)} .

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

V be a finite-dimensional representation of a finite-dimensional semisimple Lie algebra ak g over an algebraically closed field of characteristic zero. Let ak b = ak{h} ⊕ ak{n} _{+} \subset ak{g} be the Borel subalgebra determined by a choice of a Cartan subalgebra and positive roots. Let V ^{0} = \{ v \in V | ak{n} _{+(v)} = 0 \} . Then V ^{0} is an ak h -module and thus has the ak h -weight space decomposition:V ^{0} = o plus _{λ} V ^{0} _{λ} where L \subset ak{h} ^{*} . For each λ \in L , pick 0 \ne v _{λ} \in V _{λ} and V ^{λ} \subset V the ak g -submodule generated by v _{λ} and V' \subset V the ak g -submodule generated by V ^{0} . We claim: V = V' . Suppose V \ne V' . By Lie's theorem, there exists a ak{b} -weight vector in V/V' ; thus, we can find an ak{h} -weight vector v such that 0 \ne e _{i(v)} \in V' for some e _{i} among the Chevalley generators. Now, e _{i(v)} has weight \mu + \alpha _{i} . Since L is partially ordered, there is a λ \in L such that λ \ge \mu + \alpha _{i} ; i.e., λ > \mu . But this is a contradiction since λ , \mu are both primitive weights (it is known that the primitive weights are incomparable.). Similarly, each V ^{λ} is simple as a ak g -module. Indeed, if it is not simple, then, for some \mu < λ , V ^{0} _{\mu} contains some nonzero vector that is not a highest-weight vector; again a contradiction. \square

Algebraic proof 4

There is also a quick homological algebra proof; see Weibel's homological algebra book.

External links

• A blog post by Akhil Mathew

References

• Book: Hall, Brian C. . Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. 2nd. Graduate Texts in Mathematics. 222 . 2015 . Springer. 978-3319134666.
• Book: Humphreys, James E. . James E. Humphreys . Introduction to Lie Algebras and Representation Theory . Second printing, revised . Graduate Texts in Mathematics . 9 . Springer-Verlag . New York . 1973 . 0-387-90053-5 . registration .
• Book: Nathan Jacobson . Jacobson . Nathan . Lie algebras . Dover Publications, Inc. . New York . 1979 . 0-486-63832-4. Republication of the 1962 original.
• Book: Kac, Victor. Victor Kac. Infinite dimensional Lie algebras. 3rd . . 1990 . 0-521-46693-8 .
  • Book: Weibel, Charles A. . Charles Weibel . An Introduction to Homological Algebra . 1995 . Cambridge University Press .

Notes and References

1. Theorem 10.9
2. Theorem 6.11
3. Theorem 5.10
4. Theorem 4.28
5. Section 10.3
6. Section 6.2