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 [math]\displaystyle{ \mathfrak{g} }[/math] be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over [math]\displaystyle{ \mathfrak{g} }[/math] is semisimple as a module (i.e., a direct sum of simple modules.)

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 [math]\displaystyle{ \pi: \mathfrak{g} \to \mathfrak{gl}(V) }[/math], let [math]\displaystyle{ A \subset \operatorname{End}(V) }[/math] be the associative subalgebra of the endomorphism algebra of V generated by [math]\displaystyle{ \pi(\mathfrak g) }[/math]. The ring A is called the enveloping algebra of [math]\displaystyle{ \pi }[/math]. If [math]\displaystyle{ \pi }[/math] is semisimple, then A is semisimple. (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 [math]\displaystyle{ JV \subset V }[/math] implies that [math]\displaystyle{ JV = 0 }[/math]. 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 [math]\displaystyle{ \mathfrak g }[/math]-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.

Proposition — Let [math]\displaystyle{ \mathfrak g }[/math] be a semisimple finite-dimensional Lie algebra over a field of characteristic zero.

1. There exists a unique pair of elements [math]\displaystyle{ x_s, x_n }[/math] in [math]\displaystyle{ \mathfrak g }[/math] such that [math]\displaystyle{ x = x_s + x_n }[/math], [math]\displaystyle{ \operatorname{ad}(x_s) }[/math] is semisimple, [math]\displaystyle{ \operatorname{ad}(x_n) }[/math] is nilpotent and [math]\displaystyle{ [x_s, x_n] = 0 }[/math].
2. If [math]\displaystyle{ \pi : \mathfrak{g} \to \mathfrak{gl}(V) }[/math] is a finite-dimensional representation, then [math]\displaystyle{ \pi(x)_s = \pi(x_s) }[/math] and [math]\displaystyle{ \pi(x)_n = \pi(x_n) }[/math], where [math]\displaystyle{ \pi(x)_s, \pi(x)_n }[/math] denote the Jordan decomposition of the semisimple and nilpotent parts of the endomorphism [math]\displaystyle{ \pi(x) }[/math].

In short, the semisimple and nilpotent parts of an element of [math]\displaystyle{ \mathfrak g }[/math] 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 [math]\displaystyle{ \pi }[/math] is the inclusion; i.e., [math]\displaystyle{ \mathfrak g }[/math] is a subalgebra of [math]\displaystyle{ \mathfrak{gl}_n = \mathfrak{gl}(V) }[/math]. Let [math]\displaystyle{ x = S + N }[/math] be the Jordan decomposition of the endomorphism [math]\displaystyle{ x }[/math], where [math]\displaystyle{ S, N }[/math] are semisimple and nilpotent endomorphisms in [math]\displaystyle{ \mathfrak{gl}_n }[/math]. Now, [math]\displaystyle{ \operatorname{ad}_{\mathfrak{gl}_n}(x) }[/math] also has the Jordan decomposition, which can be shown (see Jordan–Chevalley decomposition) to respect the above Jordan decomposition; i.e., [math]\displaystyle{ \operatorname{ad}_{\mathfrak{gl}_n}(S), \operatorname{ad}_{\mathfrak{gl}_n}(N) }[/math] are the semisimple and nilpotent parts of [math]\displaystyle{ \operatorname{ad}_{\mathfrak{gl}_n}(x) }[/math]. Since [math]\displaystyle{ \operatorname{ad}_{\mathfrak{gl}_n}(S), \operatorname{ad}_{\mathfrak{gl}_n}(N) }[/math] are polynomials in [math]\displaystyle{ \operatorname{ad}_{\mathfrak{gl}_n}(x) }[/math] then, we see [math]\displaystyle{ \operatorname{ad}_{\mathfrak{gl}_n}(S), \operatorname{ad}_{\mathfrak{gl}_n}(N) : \mathfrak g \to \mathfrak g }[/math]. Thus, they are derivations of [math]\displaystyle{ \mathfrak{g} }[/math]. Since [math]\displaystyle{ \mathfrak{g} }[/math] is semisimple, we can find elements [math]\displaystyle{ s, n }[/math] in [math]\displaystyle{ \mathfrak{g} }[/math] such that [math]\displaystyle{ [y, S] = [y, s], y \in \mathfrak{g} }[/math] and similarly for [math]\displaystyle{ n }[/math]. Now, let A be the enveloping algebra of [math]\displaystyle{ \mathfrak{g} }[/math]; i.e., the subalgebra of the endomorphism algebra of V generated by [math]\displaystyle{ \mathfrak g }[/math]. As noted above, A has zero Jacobson radical. Since [math]\displaystyle{ [y, N - n] = 0 }[/math], we see that [math]\displaystyle{ N - n }[/math] is a nilpotent element in the center of A. But, in general, a central nilpotent belongs to the Jacobson radical; hence, [math]\displaystyle{ N = n }[/math] and thus also [math]\displaystyle{ S = s }[/math]. This proves the special case.

In general, [math]\displaystyle{ \pi(x) }[/math] is semisimple (resp. nilpotent) when [math]\displaystyle{ \operatorname{ad}(x) }[/math] is semisimple (resp. nilpotent). This immediately gives (i) and (ii). [math]\displaystyle{ \square }[/math]

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 [math]\displaystyle{ \mathfrak{g} }[/math] is the complexification of the Lie algebra of a simply connected compact Lie group [math]\displaystyle{ K }[/math]. (If, for example, [math]\displaystyle{ \mathfrak{g}=\mathrm{sl}(n;\mathbb{C}) }[/math], then [math]\displaystyle{ K=\mathrm{SU}(n) }[/math].) Given a representation [math]\displaystyle{ \pi }[/math] of [math]\displaystyle{ \mathfrak{g} }[/math] on a vector space [math]\displaystyle{ V, }[/math] one can first restrict [math]\displaystyle{ \pi }[/math] to the Lie algebra [math]\displaystyle{ \mathfrak{k} }[/math] of [math]\displaystyle{ K }[/math]. Then, since [math]\displaystyle{ K }[/math] is simply connected, there is an associated representation [math]\displaystyle{ \Pi }[/math] of [math]\displaystyle{ K }[/math]. Integration over [math]\displaystyle{ K }[/math] produces an inner product on [math]\displaystyle{ V }[/math] for which [math]\displaystyle{ \Pi }[/math] is unitary. Complete reducibility of [math]\displaystyle{ \Pi }[/math] is then immediate and elementary arguments show that the original representation [math]\displaystyle{ \pi }[/math] of [math]\displaystyle{ \mathfrak{g} }[/math] is also completely reducible.

Algebraic proof 1

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

Let [math]\displaystyle{ W \subset V }[/math] be a subrepresentation. Consider the vector subspace [math]\displaystyle{ L_W \subset \operatorname{End}(V) }[/math] that consists of all linear maps [math]\displaystyle{ t: V \to V }[/math] such that [math]\displaystyle{ t(V) \subset W }[/math] and [math]\displaystyle{ t(W) = 0 }[/math]. It has a structure of a [math]\displaystyle{ \mathfrak{g} }[/math]-module given by: for [math]\displaystyle{ x \in \mathfrak{g}, t \in L_W }[/math],

[math]\displaystyle{ x \cdot t = [\pi(x), t] }[/math].

Now, pick some projection [math]\displaystyle{ p : V \to V }[/math] onto W and consider [math]\displaystyle{ f : \mathfrak{g} \to L_W }[/math] given by [math]\displaystyle{ f(x) = [p, \pi(x)] }[/math]. Since [math]\displaystyle{ f }[/math] is a derivation, by Whitehead's lemma, we can write [math]\displaystyle{ f(x) = x \cdot t }[/math] for some [math]\displaystyle{ t \in L_W }[/math]. We then have [math]\displaystyle{ [\pi(x), p + t] = 0, x \in \mathfrak{g} }[/math]; that is to say [math]\displaystyle{ p + t }[/math] is [math]\displaystyle{ \mathfrak{g} }[/math]-linear. Also, as t kills [math]\displaystyle{ W }[/math], [math]\displaystyle{ p + t }[/math] is an idempotent such that [math]\displaystyle{ (p + t)(V) = W }[/math]. The kernel of [math]\displaystyle{ p + t }[/math] is then a complementary representation to [math]\displaystyle{ W }[/math]. [math]\displaystyle{ \square }[/math]

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 [math]\displaystyle{ C }[/math] is in the center of the universal enveloping algebra, Schur's lemma tells us that [math]\displaystyle{ C }[/math] acts as multiple [math]\displaystyle{ c_\lambda }[/math] of the identity in the irreducible representation of [math]\displaystyle{ \mathfrak{g} }[/math] with highest weight [math]\displaystyle{ \lambda }[/math]. A key point is to establish that [math]\displaystyle{ c_\lambda }[/math] is nonzero whenever the representation is nontrivial. This can be done by a general argument or by the explicit formula for [math]\displaystyle{ c_\lambda }[/math].

Consider a very special case of the theorem on complete reducibility: the case where a representation [math]\displaystyle{ V }[/math] contains a nontrivial, irreducible, invariant subspace [math]\displaystyle{ W }[/math] of codimension one. Let [math]\displaystyle{ C_V }[/math] denote the action of [math]\displaystyle{ C }[/math] on [math]\displaystyle{ V }[/math]. Since [math]\displaystyle{ V }[/math] is not irreducible, [math]\displaystyle{ C_V }[/math] is not necessarily a multiple of the identity, but it is a self-intertwining operator for [math]\displaystyle{ V }[/math]. Then the restriction of [math]\displaystyle{ C_V }[/math] to [math]\displaystyle{ W }[/math] is a nonzero multiple of the identity. But since the quotient [math]\displaystyle{ V/W }[/math] is a one dimensional—and therefore trivial—representation of [math]\displaystyle{ \mathfrak{g} }[/math], the action of [math]\displaystyle{ C }[/math] on the quotient is trivial. It then easily follows that [math]\displaystyle{ C_V }[/math] must have a nonzero kernel—and the kernel is an invariant subspace, since [math]\displaystyle{ C_V }[/math] is a self-intertwiner. The kernel is then a one-dimensional invariant subspace, whose intersection with [math]\displaystyle{ W }[/math] is zero. Thus, [math]\displaystyle{ \mathrm{ker}(V_C) }[/math] is an invariant complement to [math]\displaystyle{ W }[/math], so that [math]\displaystyle{ V }[/math] decomposes as a direct sum of irreducible subspaces:

[math]\displaystyle{ V=W\oplus\mathrm{ker}(C_V) }[/math].

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. This approach has an advantage that it can be used to weaken the finite-dimensionality assumptions (on algebra and representation).

Let [math]\displaystyle{ V }[/math] be a finite-dimensional representation of a finite-dimensional semisimple Lie algebra [math]\displaystyle{ \mathfrak g }[/math] over an algebraically closed field of characteristic zero. Let [math]\displaystyle{ \mathfrak b = \mathfrak{h} \oplus \mathfrak{n}_+ \subset \mathfrak{g} }[/math] be the Borel subalgebra determined by a choice of a Cartan subalgebra and positive roots. Let [math]\displaystyle{ V^0 = \{ v \in V | \mathfrak{n}_+(v) = 0 \} }[/math]. Then [math]\displaystyle{ V^0 }[/math] is an [math]\displaystyle{ \mathfrak h }[/math]-module and thus has the [math]\displaystyle{ \mathfrak h }[/math]-weight space decomposition:

[math]\displaystyle{ V^0 = \bigoplus_{\lambda \in L} V^0_{\lambda} }[/math]

where [math]\displaystyle{ L \subset \mathfrak{h}^* }[/math]. For each [math]\displaystyle{ \lambda \in L }[/math], pick [math]\displaystyle{ 0 \ne v_{\lambda} \in V_{\lambda} }[/math] and [math]\displaystyle{ V^{\lambda} \subset V }[/math] the [math]\displaystyle{ \mathfrak g }[/math]-submodule generated by [math]\displaystyle{ v_{\lambda} }[/math] and [math]\displaystyle{ V' \subset V }[/math] the [math]\displaystyle{ \mathfrak g }[/math]-submodule generated by [math]\displaystyle{ V^0 }[/math]. We claim: [math]\displaystyle{ V = V' }[/math]. Suppose [math]\displaystyle{ V \ne V' }[/math]. By Lie's theorem, there exists a [math]\displaystyle{ \mathfrak{b} }[/math]-weight vector in [math]\displaystyle{ V/V' }[/math]; thus, we can find an [math]\displaystyle{ \mathfrak{h} }[/math]-weight vector [math]\displaystyle{ v }[/math] such that [math]\displaystyle{ 0 \ne e_i(v) \in V' }[/math] for some [math]\displaystyle{ e_i }[/math] among the Chevalley generators. Now, [math]\displaystyle{ e_i(v) }[/math] has weight [math]\displaystyle{ \mu + \alpha_i }[/math]. Since [math]\displaystyle{ L }[/math] is partially ordered, there is a [math]\displaystyle{ \lambda \in L }[/math] such that [math]\displaystyle{ \lambda \ge \mu + \alpha_i }[/math]; i.e., [math]\displaystyle{ \lambda \gt \mu }[/math]. But this is a contradiction since [math]\displaystyle{ \lambda, \mu }[/math] are both primitive weights (it is known that the primitive weights are incomparable.). Similarly, each [math]\displaystyle{ V^{\lambda} }[/math] is simple as a [math]\displaystyle{ \mathfrak g }[/math]-module. Indeed, if it is not simple, then, for some [math]\displaystyle{ \mu \lt \lambda }[/math], [math]\displaystyle{ V^0_{\mu} }[/math] contains some nonzero vector that is not a highest-weight vector; again a contradiction. [math]\displaystyle{ \square }[/math]

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. 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. https://archive.org/details/introductiontoli00jame. 
• Lie algebras. New York: Dover Publications, Inc.. 1979. ISBN 0-486-63832-4.  Republication of the 1962 original.
• Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8. https://books.google.com/books?id=kuEjSb9teJwC&q=Victor%20G.%20Kac&pg=PP1. 
• 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.