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 gdisplaystyle mathfrak g be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over gdisplaystyle mathfrak g 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 gdisplaystyle mathfrak g is the complexification of the Lie algebra of a simply connected compact Lie group Kdisplaystyle K.^[2] (If, for example, g=sl(n;C)displaystyle mathfrak g=mathrm sl (n;mathbb C ), then K=SU(n)displaystyle K=mathrm SU (n).) Given a representation πdisplaystyle pi of gdisplaystyle mathfrak g on a vector space V,displaystyle V, we can first restrict πdisplaystyle pi to the Lie algebra kdisplaystyle mathfrak k of Kdisplaystyle K. Then, since Kdisplaystyle K is simply connected,^[3] there is an associated representation Πdisplaystyle Pi of Kdisplaystyle K. We can then use integration over Kdisplaystyle K to produce an inner product on Vdisplaystyle V for which Πdisplaystyle Pi is unitary.^[4] Complete reducibility of Πdisplaystyle Pi is then immediate and elementary arguments show that the original representation πdisplaystyle pi of gdisplaystyle mathfrak g 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 Cdisplaystyle C has in the proof. Since Cdisplaystyle C is in the center of the universal enveloping algebra, Schur's lemma tells us that Cdisplaystyle C acts as multiple cλdisplaystyle c_lambda of the identity in the irreducible representation of gdisplaystyle mathfrak g with highest weight λdisplaystyle lambda . A key point is to establish that cλdisplaystyle c_lambda is nonzero whenever the representation is nontrivial. This can be done by a general argument ^[6] or by the explicit formula for cλdisplaystyle c_lambda . We now consider a very special case of the theorem on complete reducibility: the case where a representation Vdisplaystyle V contains a nontrivial, irreducible, invariant subspace Wdisplaystyle W of codimension one. Let CVdisplaystyle C_V denote the action of Cdisplaystyle C on Vdisplaystyle V. Since Vdisplaystyle V is not irreducible, CVdisplaystyle C_V is not necessarily a multiple of the identity, but it is a self-intertwining operator for Vdisplaystyle V. Then the restriction of CVdisplaystyle C_V to Wdisplaystyle W is a nonzero multiple of the identity. But since the quotient V/Wdisplaystyle V/W is a one dimensional—and therefore trivial—representation of gdisplaystyle mathfrak g, the action of Cdisplaystyle C on the quotient is trivial. It then easily follows that CVdisplaystyle C_V must have a nonzero kernel—and the kernel is an invariant subspace, since CVdisplaystyle C_V is a self-intertwiner. The kernel is then a one-dimensional invariant subspace, whose intersection with Wdisplaystyle W is zero. Thus, ker(VC)displaystyle mathrm ker (V_C) is an invariant complement to Wdisplaystyle W, so that Vdisplaystyle V decomposes as a direct sum of irreducible subspaces:

V=W⊕ker(CV)displaystyle V=Woplus mathrm ker (C_V).

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.

External links

• A blog post by Akhil Mathew

References

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.