Elements of Cohomology for model-theorically finite-dimensional groups and Lie algebras

Abstract: We use the cohomology theory for groups invented by Hochshild and Serre to compute the first cohomology group for nilpotent groups that are definable in a finite-dimensional theory. Based on the established cohomological results, we derive some structural results : we prove a weak form of the Frattini argument for definable connected Cartan subgroups and we give a definable version of Maschke's theorem in the case of a definable abelian group without p-torsion. The same results hold when one works with Lie algebras instead of groups.