# The algebra of observables in noncommutative deformation theory

## TL;DR

This paper generalizes the Generalized Burnside Theorem for noncommutative deformation theory, proving the algebra of observables is a closure operation without field assumptions, and extends isomorphism results to broader algebra classes.

## Contribution

It extends the Generalized Burnside Theorem to cases without field assumptions and shows the observables construction acts as a closure operation for finitely generated algebras.

## Key findings

- Generalized Burnside Theorem holds over arbitrary fields.
- The algebra of observables is a closure operation.
- Isomorphism of observables persists for broader algebra classes.

## 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.