Whitehead's lemma (Lie algebras)

In algebra, Whitehead's lemma on a Lie algebra representation (named after J. H. C. Whitehead) is an important step toward the proof of Weyl's theorem on complete reducibility. Let g be a semisimple Lie algebra over a field of characteristic zero, V a finite-dimensional module over it and f : g → V a linear map such that f ( [ x , y ] ) = x f ( y ) − y f ( x ) . The lemma states that there exists a vector v in V such that f ( x ) = x v for all x.

The lemma may be interpreted in terms of Lie algebra cohomology. The proof of the lemma uses a Casimir element.