The Generalized Burnside Theorem in Noncommutative Deformation Theory

Abstract

Let A be an associative algebra over a field $k$, and let $\mathcal{M}$ be a finite family of right $A$-modules. A study of the noncommutative deformation functor $\mathrm{Def}_{\mathcal{M}}$ of the family $\mathcal{M}$ leads to the construction of the algebra $\mathcal{O}^A(\mathcal{M})$ of observables and the generalized Burnside theorem, due to Laudal (2002). In this paper, we give an overview of aspects of noncommutative deformations closely connected to the generalized Burnside theorem.