Paul Bressler
University of Haifa, Israel
On the classification of abelian extensions of Lie algebras.
Abstract: It is well-known that degree two cohomology of a Lie algebra over a field with coefficients in a module classifies abelian extensions of the Lie algebra by the module. The traditional construction of the cohomology class associated to an extension starts with a choice of a splitting of the extension which gives rise to a degree two cocycle in the Chevalley-Eilenberg complex of cochains on the Lie algebra with coefficients in the module. Lie algebras, modules and abelian extensions make sense in more general tensor categories than that of vector spaces, where splittings may not exist. In these, more general situations the standard approach does not apply, nor is the answer (degree two cohomology) correct. In my talk I will describe an alternative approach to the classification problem which, in particular, yields the correct answer in general, and which is based on explicit constructions of two mutually inverse equivalences between the category of abelian Lie algebra extensions (of a given Lie algebra by the given module) and the category of extensions of augmentation ideal in the universal enveloping algebra by the the module (in the category of modules over the Lie algebra).