Abstract

Let A be an associative algebra over a field k, and letMbe a finite family of right A-modules. A study of the noncommutative deformation functor DefM of the familyMleads to the construction of the algebra OA(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.

Highlights

  • Let k be a field and let A be an associative k-algebra

  • For any right A-module M, there is a commutative deformation functor DefM : l → Sets defined on the category l of local Artinian commutative k-algebras with residue field k

  • DefM has a pro-representing hull or a formal moduli (H, MH ); see Laudal [2, Theorem 3.1]. This means that H is a complete r-pointed k-algebra in the pro-categoryar, and that MH ∈ DefM(H) is a family defined over H with the following versal property: for any algebra R in ar and any deformation MR ∈ DefM(R), there is a homomorphism φ : H → R such that DefM(φ)(MH ) = MR

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Summary

Introduction

For any right A-module M , there is a commutative deformation functor DefM : l → Sets defined on the category l of local Artinian commutative k-algebras with residue field k. We recall that for an algebra R in l, a deformation of M to R is a pair (MR, τ ), where MR is an R-A bimodule (on which k acts centrally) that is R-flat, and τ : k ⊗R MR → M is an isomorphism of right A-modules. A deformation of M to R is defined to be a pair (MR, {τi}1≤i≤r), where MR is an R-A bimodule (on which k acts centrally) that is R-flat, and τi : ki ⊗R MR → Mi is an isomorphism of right A-modules for 1 ≤ i ≤ r. This means that the family M has exactly the same module-theoretic properties, in terms of (higher) extensions and Massey products, considered as a family of modules over B as over A

Noncommutative deformations of modules
The generalized Burnside theorem
Properties of the algebra of observables
Examples: representations of ordered sets
A hereditary example
The diamond
Full Text
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