Whitehead's lemma

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 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).