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).
be a semisimple Lie algebra over a field of characteristic zero.
The theorem states that every finite-dimensional module over
)[1] 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
be the associative subalgebra of the endomorphism algebra of V generated by
[2] (Proof: Since A is a finite-dimensional algebra, it is an Artinian ring; in particular, the Jacobson radical J is nilpotent.
(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.)
be a semisimple finite-dimensional Lie algebra over a field of characteristic zero and
[a] In short, the semisimple and nilpotent parts of an element of
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
But, in general, a central nilpotent belongs to the Jacobson radical; hence,
[clarification needed] This immediately gives (i) and (ii).
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 simply connected,[5] there is an associated representation
is then immediate and elementary arguments show that the original representation
The theorem is an easy consequence of Whitehead's lemma, which says
The proof is essentially due to Whitehead.
is a derivation, by Whitehead's lemma, we can write
Whitehead's lemma is typically proved by means of the quadratic Casimir element of the universal enveloping algebra,[8] and there is also a proof of the theorem that uses the Casimir element directly instead of Whitehead's lemma.
is in the center of the universal enveloping algebra, Schur's lemma tells us that
This can be done by a general argument [9] or by the explicit formula for
contains a nontrivial, irreducible, invariant subspace
is not necessarily a multiple of the identity, but it is a self-intertwining operator for
must have a nonzero kernel—and the kernel is an invariant subspace, since
The kernel is then a one-dimensional invariant subspace, whose intersection with
decomposes as a direct sum of irreducible subspaces: Although this establishes only a very special case of the desired result, this step is actually the critical one in the general argument.
[10] This approach has an advantage that it can be used to weaken the finite-dimensionality assumptions (on algebra and representation).