Additive combinatorics methods in associative algebras
Vincent Beck (MAPMO), C\'edric Lecouvey (LMPT)

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.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Topics in Algebra
