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In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of
Lie algebra representation In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is g ...
s (specifically in the
representation theory of semisimple Lie algebras In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is a ...
). Let \mathfrak be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over \mathfrak is
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
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 mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself ...
in the following way. Given a finite-dimensional Lie algebra representation \pi: \mathfrak \to \mathfrak(V), let A \subset \operatorname(V) be the associative subalgebra of the endomorphism algebra of ''V'' generated by \pi(\mathfrak g). The ring ''A'' is called the enveloping algebra of \pi. If \pi 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 JV \subset V implies that JV = 0. 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 \mathfrak g-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. ''Proof'': First we prove the special case of (i) and (ii) when \pi is the inclusion; i.e., \mathfrak g is a subalgebra of \mathfrak_n = \mathfrak(V). Let x = S + N be the Jordan decomposition of the endomorphism x, where S, N are semisimple and nilpotent endomorphisms in \mathfrak_n. Now, \operatorname_(x) also has the Jordan decomposition, which can be shown (see Jordan–Chevalley decomposition#Lie algebras) to respect the above Jordan decomposition; i.e., \operatorname_(S), \operatorname_(N) are the semisimple and nilpotent parts of \operatorname_(x). Since \operatorname_(S), \operatorname_(N) are polynomials in \operatorname_(x) then, we see \operatorname_(S), \operatorname_(N) : \mathfrak g \to \mathfrak g. Thus, they are derivations of \mathfrak. Since \mathfrak is semisimple, we can find elements s, n in \mathfrak such that , S= , s y \in \mathfrak and similarly for n. Now, let ''A'' be the enveloping algebra of \mathfrak; i.e., the subalgebra of the endomorphism algebra of ''V'' generated by \mathfrak g. As noted above, ''A'' has zero Jacobson radical. Since , N - n= 0, we see that N - n is a nilpotent element in the center of ''A''. But, in general, a central nilpotent belongs to the Jacobson radical; hence, N = n and thus also S = s. This proves the special case. In general, \pi(x) is semisimple (resp. nilpotent) when \operatorname(x) is semisimple (resp. nilpotent). This immediately gives (i) and (ii). \square


Proofs


Analytic proof

Weyl's original proof (for complex semisimple Lie algebras) was analytic in nature: it famously used the
unitarian trick In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some g ...
. Specifically, one can show that every complex semisimple Lie algebra \mathfrak is the complexification of the Lie algebra of a simply connected compact Lie group K. (If, for example, \mathfrak=\mathrm(n;\mathbb), then K=\mathrm(n).) Given a representation \pi of \mathfrak on a vector space V, one can first restrict \pi to the Lie algebra \mathfrak 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 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 an easy consequence of Whitehead's lemma, which says V \to \operatorname(\mathfrak g, V), v \mapsto \cdot v is surjective, where a linear map f: \mathfrak g \to V is a
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
if f(
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= 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 L_W \subset \operatorname(V) that consists of all linear maps t: V \to V such that t(V) \subset W and t(W) = 0. It has a structure of a \mathfrak-module given by: for x \in \mathfrak, t \in L_W, :x \cdot t =
pi(x), t The number (; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number appears in many formulas across mathematics and physics. It is an irratio ...
/math>. Now, pick some projection p : V \to V onto ''W'' and consider f : \mathfrak \to L_W given by f(x) = , \pi(x)/math>. Since f is a derivation, by Whitehead's lemma, we can write f(x) = x \cdot t for some t \in L_W. We then have pi(x), p + t= 0, x \in \mathfrak; that is to say p + t is \mathfrak-linear. Also, as ''t'' kills W, p + t is an idempotent such that (p + t)(V) = W. The kernel of p + t is then a complementary representation to W. \square See also Weibel's
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
book.


Algebraic proof 2

Whitehead's lemma is typically proved by means of the quadratic Casimir element of the
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representati ...
, 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 In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ' ...
tells us that C acts as multiple c_\lambda of the identity in the irreducible representation of \mathfrak 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, 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(V_C) is an invariant complement to W, so that V decomposes as a direct sum of irreducible subspaces: :V=W\oplus\mathrm(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.


Algebraic proof 3

The theorem can be deduced from the theory of
Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...
s, which characterizes a simple module as a quotient of a Verma module by a
maximal submodule In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
. This approach has an advantage that it can be used to weaken the finite-dimensionality assumptions (on algebra and representation). Let V be a finite-dimensional representation of a finite-dimensional semisimple Lie algebra \mathfrak g over an algebraically closed field of characteristic zero. Let \mathfrak b = \mathfrak \oplus \mathfrak_+ \subset \mathfrak be the
Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie group, ...
determined by a choice of a Cartan subalgebra and positive roots. Let V^0 = \. Then V^0 is an \mathfrak h-module and thus has the \mathfrak h-weight space decomposition: :V^0 = \bigoplus_ V^0_ where L \subset \mathfrak^*. For each \lambda \in L, pick 0 \ne v_ \in V_ and V^ \subset V the \mathfrak g-submodule generated by v_ and V' \subset V the \mathfrak g-submodule generated by V^0. We claim: V = V'. Suppose V \ne V'. By
Lie's theorem In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if \pi: \mathfrak \to \mathfrak(V) is a finite-dimensional representation of a solvable Lie algebra, the ...
, there exists a \mathfrak-weight vector in V/V'; thus, we can find an \mathfrak-weight vector v such that 0 \ne e_i(v) \in V' for some e_i among the
Chevalley generators In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra ...
. Now, e_i(v) has weight \mu + \alpha_i. Since L is partially ordered, there is a \lambda \in L such that \lambda \ge \mu + \alpha_i; i.e., \lambda > \mu. But this is a contradiction since \lambda, \mu are both primitive weights (it is known that the primitive weights are incomparable.). Similarly, each V^ is simple as a \mathfrak g-module. Indeed, if it is not simple, then, for some \mu < \lambda, V^0_ contains some nonzero vector that is not a highest-weight vector; again a contradiction. \square


External links

*
blog post
by Akhil Mathew


References

* * * Jacobson, Nathan, ''Lie algebras'', Republication of the 1962 original. Dover Publications, Inc., New York, 1979. * * * {{cite book , last=Weibel , first=Charles A. , author-link=Charles Weibel , title=An Introduction to Homological Algebra , year=1995 , publisher=Cambridge University Press Lie algebras