In homological algebra, Whitehead's lemmas (named after J. H. C. Whitehead) 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.
[1] 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.
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
In terms of Lie algebra cohomology, this is, by definition, equivalent to the fact that
[2] 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
act nontrivially, so
be a finite-dimensional semisimple Lie algebra over a field of characteristic zero and
a finite-dimensional representation (which is semisimple but the proof does not use that fact).
= ker ( π ) ⊕
is semisimple, the trace form
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
, it acts on V as a linear endomorphism (namely,
The key property is that it commutes with
tr ( π ( c ) ) = ∑ tr ( π (
Now, by Fitting's lemma, we have the vector space decomposition
is a (well-defined) nilpotent endomorphism for
Hence, it is enough to prove the lemma separately for
{\displaystyle \dim({\mathfrak {g}}/\operatorname {ker} (\pi ))=\operatorname {tr} (\pi (c))=0}
is a trivial representation.
For notational simplicity, we will drop
denote the trace form used earlier.
, the second term of the expansion of
has the required property.