Diagrams of Lie algebras

Abstract

Diagrams of Lie algebras have natural cohomology and deformation theories. The relationship between Lie and Hochschild cohomology allows one to reduce these to the associative case, where the cohomology comparison theorem asserts that for every diagram of associative algebras there is a single associative algebra whose cohomology and deformation theories are the same as those of the entire diagram. We show that the cohomology of a diagram of Lie algebras with coefficients in a diagram of Lie modules is canonically isomorphic to that of a single associative algebra with coefficients in a single bimodule, but the existence of operations analogous to those in the associative case remains unknown. In the last section, we construct a single cochain complex from a diagram of cochain complexes and derive the usual mapping cone and cylinder.