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 ${\mathfrak {g}}$ 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.)

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 ${\mathfrak {g}}$ is the complexification of the Lie algebra of a simply connected compact Lie group $K$. (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 $V,$ one can first restrict $\pi$ to the Lie algebra ${\mathfrak {k}}$ of $K$. Then, since $K$ is simply connected, there is an associated representation $\Pi$ of $K$. Integration over $K$ produces an inner product on $V$ for which $\Pi$ is unitary. Complete reducibility of $\Pi$ is then immediate and elementary arguments show that the original representation $\pi$ of ${\mathfrak {g}}$ is also completely reducible.

Algebraic proof 1

Let $(\pi ,V)$ be a finite-dimensional representation of a Lie algebra ${\mathfrak {g}}$ over a field of characteristic zero. The theorem is a (slightly tricky) consequence of Whitehead's lemma, which says $V\to \operatorname {Der} ({\mathfrak {g}},V),v\mapsto \cdot v$ is surjective, where a linear map $f:{\mathfrak {g}}\to V$ is a derivation if $f([x,y])=x\cdot f(y)-y\cdot f(x)$. The proof is essentially due to Whitehead.

Let $W\subset V$ be a subrepresentation. Consider the vector subspace ${\mathfrak {t}}_{W}\subset \operatorname {End} (V)$ that consists of all linear maps $t:V\to V$ such that $t(V)\subset W$ and $t(W)=0$. Clearly, ${\mathfrak {t}}_{W}$ is a Lie subalgebra of ${\mathfrak {gl}}(V)$. Moreover, it has a structure of a ${\mathfrak {g}}$-module given by: for $x\in {\mathfrak {g}},t\in {\mathfrak {t}}_{W}$,

$x\cdot t=[\pi (x),t]$.

Now, pick some projection $p:V\to V$ onto W and consider $f:{\mathfrak {g}}\to {\mathfrak {t}}_{W}$ given by $f(x)=[p,\pi (x)]$. Since $f$ is then a derivation, we can write $f(x)=x\cdot t$ for some $t\in {\mathfrak {t}}_{W}$. We then have $[\pi (x),p+t]=0,x\in {\mathfrak {g}}$; that is to say $p+t$ is ${\mathfrak {g}}$-linear. The kernel of $1-(p+t)$ is then the complementary representation to $W$. $\square$

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, 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 $C$ 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 or by the explicit formula for $c_{\lambda }$.

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 this establishes 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, 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, 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, vol. 140 (2nd ed.), Boston: Birkhäuser, ISBN 0-8176-4259-5
• Weibel, Charles A. (1995). An Introduction to Homological Algebra. Cambridge University Press.