The algebra of observables in noncommutative deformation theory

Abstract

We consider the algebra O(M) of observables and the (formally) versal morphism η : A → O(M) defined by the noncommutative deformation functor DefM of a family M = {M1, . . . , Mr} of right modules over an associative k-algebra A. By the Generalized Burnside Theorem, due to Laudal, η is an isomorphism when A is finite dimensional, M is the family of simple A-modules, and k is an algebraically closed field. The purpose of this paper is twofold: First, we prove a form of the Generalized Burnside Theorem that is more general, where there is no assumption on the field k. Secondly, we prove that the O-construction is a closure operation when A is any finitely generated k-algebra and M is any family of finite dimensional A-modules, in the sense that ηB : B → OB(M) is an isomorphism when B = O(M) and M is considered as a family of B-modules.