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

The enveloping algebra is semisimple

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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  , let   be the associative subalgebra of the endomorphism algebra of V generated by  . The ring A is called the enveloping algebra of  . If   is semisimple, then A is semisimple.[2] (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   implies that  . 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  -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

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Here is a typical application.[3]

Proposition — Let   be a semisimple finite-dimensional Lie algebra over a field of characteristic zero.[a]

  1. There exists a unique pair of elements   in   such that  ,   is semisimple,   is nilpotent and  .
  2. If   is a finite-dimensional representation, then   and  , where   denote the Jordan decomposition of the semisimple and nilpotent parts of the endomorphism  .

In short, the semisimple and nilpotent parts of an element of   are well-defined and are determined independent of a faithful finite-dimensional representation.

Proof: First we prove the special case of (i) and (ii) when   is the inclusion; i.e.,   is a subalgebra of  . Let   be the Jordan decomposition of the endomorphism  , where   are semisimple and nilpotent endomorphisms in  . Now,   also has the Jordan decomposition, which can be shown (see Jordan–Chevalley decomposition) to respect the above Jordan decomposition; i.e.,   are the semisimple and nilpotent parts of  . Since   are polynomials in   then, we see  . Thus, they are derivations of  . Since   is semisimple, we can find elements   in   such that   and similarly for  . Now, let A be the enveloping algebra of  ; i.e., the subalgebra of the endomorphism algebra of V generated by  . As noted above, A has zero Jacobson radical. Since  , we see that   is a nilpotent element in the center of A. But, in general, a central nilpotent belongs to the Jacobson radical; hence,   and thus also  . This proves the special case.

In general,   is semisimple (resp. nilpotent) when   is semisimple (resp. nilpotent).[clarification needed] This immediately gives (i) and (ii).  

Proofs

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Analytic proof

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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   is the complexification of the Lie algebra of a simply connected compact Lie group  .[4] (If, for example,  , then  .) Given a representation   of   on a vector space   one can first restrict   to the Lie algebra   of  . Then, since   is simply connected,[5] there is an associated representation   of  . Integration over   produces an inner product on   for which   is unitary.[6] Complete reducibility of   is then immediate and elementary arguments show that the original representation   of   is also completely reducible.

Algebraic proof 1

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Let   be a finite-dimensional representation of a Lie algebra   over a field of characteristic zero. The theorem is an easy consequence of Whitehead's lemma, which says   is surjective, where a linear map   is a derivation if  . The proof is essentially due to Whitehead.[7]

Let   be a subrepresentation. Consider the vector subspace   that consists of all linear maps   such that   and  . It has a structure of a  -module given by: for  ,

 .

Now, pick some projection   onto W and consider   given by  . Since   is a derivation, by Whitehead's lemma, we can write   for some  . We then have  ; that is to say   is  -linear. Also, as t kills  ,   is an idempotent such that  . The kernel of   is then a complementary representation to  .  

Algebraic proof 2

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Whitehead's lemma is typically proved by means of the quadratic Casimir element of the universal enveloping algebra,[8] 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   is in the center of the universal enveloping algebra, Schur's lemma tells us that   acts as multiple   of the identity in the irreducible representation of   with highest weight  . A key point is to establish that   is nonzero whenever the representation is nontrivial. This can be done by a general argument [9] or by the explicit formula for  .

Consider a very special case of the theorem on complete reducibility: the case where a representation   contains a nontrivial, irreducible, invariant subspace   of codimension one. Let   denote the action of   on  . Since   is not irreducible,   is not necessarily a multiple of the identity, but it is a self-intertwining operator for  . Then the restriction of   to   is a nonzero multiple of the identity. But since the quotient   is a one dimensional—and therefore trivial—representation of  , the action of   on the quotient is trivial. It then easily follows that   must have a nonzero kernel—and the kernel is an invariant subspace, since   is a self-intertwiner. The kernel is then a one-dimensional invariant subspace, whose intersection with   is zero. Thus,   is an invariant complement to  , so that   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

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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.[10] This approach has an advantage that it can be used to weaken the finite-dimensionality assumptions (on algebra and representation).

Let   be a finite-dimensional representation of a finite-dimensional semisimple Lie algebra   over an algebraically closed field of characteristic zero. Let   be the Borel subalgebra determined by a choice of a Cartan subalgebra and positive roots. Let  . Then   is an  -module and thus has the  -weight space decomposition:

 

where  . For each  , pick   and   the  -submodule generated by   and   the  -submodule generated by  . We claim:  . Suppose  . By Lie's theorem, there exists a  -weight vector in  ; thus, we can find an  -weight vector   such that   for some   among the Chevalley generators. Now,   has weight  . Since   is partially ordered, there is a   such that  ; i.e.,  . But this is a contradiction since   are both primitive weights (it is known that the primitive weights are incomparable.[clarification needed]). Similarly, each   is simple as a  -module. Indeed, if it is not simple, then, for some  ,   contains some nonzero vector that is not a highest-weight vector; again a contradiction.[clarification needed]  

Algebraic proof 4

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

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References

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  1. ^ Editorial note: this fact is usually stated for a field of characteristic zero, but the proof needs only that the base field be perfect.
  1. ^ Hall 2015 Theorem 10.9
  2. ^ Jacobson 1979, Ch. II, § 5, Theorem 10.
  3. ^ Jacobson 1979, Ch. III, § 11, Theorem 17.
  4. ^ Knapp 2002 Theorem 6.11
  5. ^ Hall 2015 Theorem 5.10
  6. ^ Hall 2015 Theorem 4.28
  7. ^ Jacobson 1979, Ch. III, § 7.
  8. ^ Hall 2015 Section 10.3
  9. ^ Humphreys 1973 Section 6.2
  10. ^ Kac 1990, Lemma 9.5.
  • Hall, Brian C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Vol. 222 (2nd ed.). Springer. ISBN 978-3319134666.
  • Humphreys, James E. (1973). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. Vol. 9 (Second printing, revised ed.). New York: Springer-Verlag. ISBN 0-387-90053-5.
  • Jacobson, Nathan (1979). Lie algebras. New York: Dover Publications, Inc. ISBN 0-486-63832-4. Republication of the 1962 original.
  • Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8.
  • Knapp, Anthony W. (2002), Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140 (2nd ed.), Boston: Birkhäuser, ISBN 0-8176-4259-5
  • Weibel, Charles A. (1995). An Introduction to Homological Algebra. Cambridge University Press.