Weyl's theorem on complete reducibility

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In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations. 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]

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 .[2] (If, for example, , then .) Given a representation of on a vector space we can first restrict to the Lie algebra of . Then, since is simply connected,[3] there is an associated representation of . We can then use integration over to produce an inner product on for which is unitary.[4] Complete reducibility of is then immediate and elementary arguments show that the original representation of is also completely reducible.

The usual algebraic proof makes use of the quadratic Casimir element of the universal enveloping algebra,[5] and can be seen as a consequence of Whitehead's lemma (see Weibel's homological algebra book).

We now explain briefly the role that the quadratic Casimir element has in the proof. Since 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 [6] or by the explicit formula for . We now 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 we have so far established only a very special case of the desired result, this step is actually the critical one in the general argument.

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References[edit]

  1. ^ Hall 2015 Theorem 10.9
  2. ^ Knapp 2002 Theorem 6.11
  3. ^ Hall 2015 Theorem 5.10
  4. ^ Hall 2015 Theorem 4.28
  5. ^ Hall 2015 Section 10.3
  6. ^ Humphreys 1973 Section 6.2
  • 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
  • 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.