Relative Lie algebra cohomology
Let \((\mathfrak g,\mathfrak f)\) be a pair of finite dimensional Lie algebras where \(\mathfrak f\) is a Lie subalgebra of \(\mathfrak g\), and let \((U\mathfrak g,U\mathfrak f)\) be the corresponding pair of universal enveloping algebras. The author studies the Hochschild relative cohomology groups and the Hochschild complex [see \textit{G. Hochschild}, Trans. Am. Math. Soc. 82, 246--269 (1956; Zbl 0070.26903)]. It is known that if the Hochschild complex has a U\({\mathfrak f}\)-homotopy then the relative cohomology groups \(H^*({\mathfrak g},{\mathfrak f},V)\) and \(H^*(U{\mathfrak g},U{\mathfrak f},V)\) are identical. The purpose of the paper is to investigate the existence of a U\({\mathfrak f}\)-homotopy and the possible isomorphism of the two relative cohomology groups. It is shown that the relative cohomology groups are related by a spectral sequence and some good conditions are given for the isomorphism. A vanishing property for the Hochschild relative cohomology groups is obtained and it is shown that in general the two cohomology groups are not isomorphic.
scientific article; zbMATH DE number 4116773
scientific article