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

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

Here is a typical application.[3]

Proposition  Let be a semisimple finite-dimensional Lie algebra over a field of characteristic zero.[4]

  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#Lie algebras) 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). This immediately gives (i) and (ii).

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 is the complexification of the Lie algebra of a simply connected compact Lie group .[5] (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,[6] there is an associated representation of . Integration over produces an inner product on for which is unitary.[7] Complete reducibility of is then immediate and elementary arguments show that the original representation of is also completely reducible.

Algebraic proof 1

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.[8]

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 .

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,[9] 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 [10] 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

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.[11] 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.). 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.

References

  1. Hall 2015 Theorem 10.9
  2. Jacobson 1962, Ch. II, § 5, Theorem 10.
  3. Jacobson 1962, Ch. III, § 11, Theorem 17.
  4. Editorial note: this fact is usually stated for a field of characteristic zero, but the proof needs only that the base field be perfect.
  5. Knapp 2002 Theorem 6.11
  6. Hall 2015 Theorem 5.10
  7. Hall 2015 Theorem 4.28
  8. Jacobson 1961, Ch. III, § 7.
  9. Hall 2015 Section 10.3
  10. Humphreys 1973 Section 6.2
  11. Kac 1990, Lemma 9.5.
  • Hall, Brian C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. 222 (2nd ed.). Springer. ISBN 978-3319134666.
  • Humphreys, James E. (1973). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. 9 (Second printing, revised ed.). New York: Springer-Verlag. ISBN 0-387-90053-5.
  • Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
  • 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, 140 (2nd ed.), Boston: Birkhäuser, ISBN 0-8176-4259-5
  • Weibel, Charles A. (1995). An Introduction to Homological Algebra. Cambridge University Press.
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