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. Let ${\mathfrak {g}}$ be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ is the complexification of the Lie algebra of a simply connected compact Lie group $K$ .^[2] (If, for example, ${\mathfrak {g}}=\mathrm {sl} (n;\mathbb {C} )$ , then $K=\mathrm {SU} (n)$ .) Given a representation $\pi$ of ${\mathfrak {g}}$ on a vector space $V,$ we can first restrict $\pi$ to the Lie algebra $\mathfrak{k}$ of $K$ . Then, since $K$ is simply connected,^[3] there is an associated representation $\Pi$ of $K$ . We can then use integration over $K$ to produce 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 ${\mathfrak {g}}$ 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 $C$ has in the proof. Since $C$ is in the center of the universal enveloping algebra, Schur's lemma tells us that $C$ acts as multiple $c_{\lambda }$ of the identity in the irreducible representation of ${\mathfrak {g}}$ with highest weight $\lambda$ . A key point is to establish that $c_{\lambda }$ is nonzero whenever the representation is nontrivial. This can be done by a general argument ^[6] or by the explicit formula for $c_{\lambda }$ . We now 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 ${\mathfrak {g}}$ , 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, $\mathrm {ker} (V_{C})$ is an invariant complement to $W$ , so that $V$ decomposes as a direct sum of irreducible subspaces:

V=W\oplus \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

↑ Hall 2015 Theorem 10.9
↑ Knapp 2002 Theorem 6.11
↑ Hall 2015 Theorem 5.10
↑ Hall 2015 Theorem 4.28
↑ Hall 2015 Section 10.3
↑ 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.

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[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