Whitehead's lemma (Lie algebras)

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. 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 [math]\displaystyle{ \mathfrak{g} }[/math] be a finite-dimensional, semisimple Lie algebra over a field of characteristic zero, V a finite-dimensional module over it, and [math]\displaystyle{ f\colon \mathfrak{g} \to V }[/math] a linear map such that

[math]\displaystyle{ f([x, y]) = xf(y) - yf(x) }[/math].

Then there exists a vector [math]\displaystyle{ v \in V }[/math] such that [math]\displaystyle{ f(x) = xv }[/math] for all [math]\displaystyle{ x \in \mathfrak{g} }[/math]. In terms of Lie algebra cohomology, this is, by definition, equivalent to the fact that [math]\displaystyle{ H^1(\mathfrak{g},V) = 0 }[/math] 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 [math]\displaystyle{ H^2(\mathfrak{g},V) = 0 }[/math].

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 [math]\displaystyle{ V }[/math] be irreducible under the [math]\displaystyle{ \mathfrak{g} }[/math]-action and let [math]\displaystyle{ \mathfrak{g} }[/math] act nontrivially, so [math]\displaystyle{ \mathfrak{g} \cdot V \neq 0 }[/math]. Then [math]\displaystyle{ H^q(\mathfrak{g},V) = 0 }[/math] for all [math]\displaystyle{ q \geq 0 }[/math].

References

• Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4