In
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, 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
be a
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ...
over a
field of
characteristic zero. The theorem states that every finite-dimensional module over
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
, let
be the associative subalgebra of the endomorphism algebra of ''V'' generated by
. The ring ''A'' is called the enveloping algebra of
. If
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
implies that
. 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
-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
is the inclusion; i.e.,
is a subalgebra of
. Let
be the Jordan decomposition of the endomorphism
, where
are semisimple and nilpotent endomorphisms in
. Now,
also has the Jordan decomposition, which can be shown (see
Jordan–Chevalley decomposition) to respect the above Jordan decomposition; i.e.,
are the semisimple and nilpotent parts of
. Since
are polynomials in
then, we see
. Thus, they are derivations of
. Since
is semisimple, we can find elements
in
such that
and similarly for
. Now, let ''A'' be the enveloping algebra of
; i.e., the subalgebra of the endomorphism algebra of ''V'' generated by
. As noted above, ''A'' has zero Jacobson radical. Since
, we see that
is a nilpotent element in the center of ''A''. But, in general, a central nilpotent belongs to the Jacobson radical; hence,
and thus also
. This proves the special case.
In general,
is semisimple (resp. nilpotent) when
is semisimple (resp. nilpotent). This immediately gives (i) and (ii).
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
is the complexification of the Lie algebra of a simply connected compact
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
. (If, for example,
, then
.) Given a representation
of
on a vector space
one can first restrict
to the Lie algebra
of
. Then,
since is simply connected, there is an associated representation
of
. Integration over
produces an inner product on
for which
is unitary. Complete reducibility of
is then immediate and elementary arguments show that the original representation
of
is also completely reducible.
Algebraic proof 1
Let
be a finite-dimensional representation of a Lie algebra
over a field of characteristic zero. The theorem is an easy consequence of
Whitehead's lemma, which says
is surjective, where a linear map
is a
derivation if
. The proof is essentially due to Whitehead.
Let
be a subrepresentation. Consider the vector subspace
that consists of all linear maps
such that
and
. It has a structure of a
-module given by: for
,
: