Additive combinatorics methods in associative algebras

TL;DR

This paper extends additive combinatorics techniques from groups to associative algebras, establishing analogues of key theorems and classifying certain finite-dimensional algebras, with applications to monoids.

Contribution

It introduces a linear framework for additive combinatorics in associative algebras, including analogues of classical theorems and a classification of finite-dimensional algebras with finitely many subalgebras.

Findings

Established algebraic analogues of Diderrich-Kneser and Hamidoune theorems

Classified finite-dimensional algebras with finitely many subalgebras over infinite fields

Derived lower bounds for Minkowski products in finite monoids

Abstract

We adapt methods coming from additive combinatorics in groups to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Diderrich-Kneser's and Hamidoune's theorems on sumsets and Tao's theorem on sets of small doubling. In passing we classify the finite-dimensional algebras over infinite fields with finitely many subalgebras. These algebras play a crucial role in our linear version of Diderrich-Kneser's theorem. We also explain how the original theorems for groups we linearize can be easily deduced from our results applied to group algebras. Finally, we give lower bounds for the Minkowski product of two subsets in finite monoids by using their associated monoid algebras.