Whitehead's lemma (Lie algebra)


In homological algebra, Whitehead's lemmas represent a series of statements regarding representation theory of finite-dimensional, semisimple Lie algebras in characteristic zero. Historically, they are regarded as leading to the discovery of Lie algebra cohomology.
One usually makes the distinction between Whitehead's first and second lemma for the corresponding statements about first and second order cohomology, respectively, but there are similar statements pertaining to Lie algebra cohomology in arbitrary orders which are also attributed to Whitehead.
The first Whitehead lemma is an important step toward the proof of Weyl's theorem on complete reducibility.

Statements

Without mentioning cohomology groups, one can state Whitehead's first lemma as follows: Let be a finite-dimensional, semisimple Lie algebra over a field of characteristic zero, V a finite-dimensional module over it, and a linear map such that
Then there exists a vector such that for all.
In terms of Lie algebra cohomology, this is, by definition, equivalent to the fact that for every such representation. The proof uses a Casimir element.
Similarly, Whitehead's second lemma states that under the conditions of the first lemma, also.
Another related statement, which is also attributed to Whitehead, describes Lie algebra cohomology in arbitrary order: Given the same conditions as in the previous two statements, but further let be irreducible under the -action and let act nontrivially, so. Then for all.

Proof

As above, let be a finite-dimensional semisimple Lie algebra over a field of characteristic zero and a finite-dimensional representation.
Let where is an ideal of. Then, since is semisimple, the trace form, relative to, is nondegenerate on. Let be a basis of and the dual basis with respect to this trace form. Then define the Casimir element by
which is an element of the universal enveloping algebra of. Via, it acts on V as a linear endomorphism The key property is that it commutes with in the sense for each element. Also,
Now, by Fitting's lemma, we have the vector space decomposition such that is a nilpotent endomorphism for and is an automorphism for. Since commutes with, each is a -submodule. Hence, it is enough to prove the lemma separately for and.
First, suppose is a nilpotent endomorphism. Then, by the early observation, ; that is, is a trivial representation. Since, the condition on implies that for each ; i.e., the zero vector satisfies the requirement.
Second, suppose is an automorphism. For notational simplicity, we will drop and write. Also let denote the trace form used earlier. Let, which is a vector in. Then
Now,
and, since, the second term of the expansion of is
Thus,
Since is invertible and commutes with, the vector has the required property.