Cohomology of Lie Algebras

Abstract

In this Chapter we shall give a further application of the theory of derived functors. Starting with a Lie algebra g over the field K, we pass to the universal enveloping algebra Ug and define cohomology groups H ⁿ(g A) for every (left) g-module A, by regarding A as a Ug-module. In Sections 1 through 4 we will proceed in a way parallel to that adopted in Chapter VI in presenting the cohomology theory of groups. We therefore allow ourselves in those sections to leave most of the proofs to the reader. Since our primary concern is with the homological aspects of Lie algebra theory, we will not give proofs of two deep results of Lie algebra theory although they are fundamental for the development of the cohomology theory of Lie algebras; namely, we shall not give a proof for the Birkhoff-Witt Theorem (Theorem 1.2) nor of Theorem 5.2 which says that the bilinear form of certain representations of semisimple Lie algebras is non-degenerate. Proofs of both results are easily accessible in the literature.