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.
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 (see the proof below).[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
be irreducible under the
-action and let
act nontrivially, so
. Then
for all
.[3]
As above, let
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).
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
![{\displaystyle c=\sum _{i}e_{i}e^{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eccf71c8bc5b109ab71ef6d42a0cc18cb0e32210)
which is an element of the universal enveloping algebra of
. Via
, it acts on V as a linear endomorphism (namely,
.) 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 (well-defined) 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
![{\displaystyle xw=\sum _{i}e_{i}xf(e^{i})+\sum _{i}[x,e_{i}]f(e^{i}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d5f94f3880585a5e43af9df247d4834d6ed334)
Now,
![{\displaystyle [x,e_{i}]=\sum _{j}([x,e_{i}],e^{j})e_{j}=-\sum _{j}([x,e^{j}],e_{i})e_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67ce0a116d3ed86f32452bcdf598ad66d1a3c0d8)
and, since
, the second term of the expansion of
is
![{\displaystyle -\sum _{j}e_{j}f([x,e^{j}])=-\sum _{i}e_{i}(xf(e^{i})-e^{i}f(x)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48d3abb68786e5c053ea886a6a48144c51f2ded4)
Thus,
![{\displaystyle xw=\sum _{i}e_{i}e^{i}f(x)=cf(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22b5f63bb99087e1a8d4d10dca7130095c9da0aa)
Since
is invertible and
commutes with
, the vector
has the required property.
Notes
References