The algebra of observables in noncommutative deformation theory

Abstract

We consider the algebra $\mathcal{O}(\mathsf{M})$ of observables and the (formally) versal morphism $\eta :A\to \mathcal{O}(\mathsf{M})$ defined by the noncommutative deformation functor ${\mathsf{Def}}_{\mathsf{M}}$ of a family $\mathsf{M}=\{M_1,\dots ,M_r\}$ of right modules over an associative $k$-algebra $A$. By the Generalized Burnside Theorem, due to Laudal, $\eta$ is an isomorphism when $A$ is finite dimensional, $\mathsf{M}$ 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 $\mathcal{O}$-construction is a closure operation when $A$ is any finitely generated $k$-algebra and $\mathsf{M}$ is any family of finite dimensional $A$-modules, in the sense that ${\eta }_B:B\to {\mathcal{O}}^B(\mathsf{M})$ is an isomorphism when $B=\mathcal{O}(\mathsf{M})$ and $\mathsf{M}$ is considered as a family of $B$-modules.

9 pages