Growth, entropy and commutativity of algebras satisfying prescribed relations

TL;DR

This paper investigates the growth, entropy, and commutativity of algebras defined by specific relations, extending Golod-Shafarevich theory, constructing examples with various growth behaviors, and answering open questions in algebraic structure and noncommutative singularities.

Contribution

It provides new bounds on growth functions of Golod-Shafarevich algebras, constructs algebras with prescribed subexponential growth, and addresses open questions on algebraic commutativity and nil algebras.

Findings

Bounds for growth functions based on relations

Construction of algebras with subexponential growth

Answering open questions on algebraic commutativity

Abstract

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod-Shafarevich algebras. This paper provides bounds for the growth function on images of Golod-Shafarevich algebras based upon the number of defining relations. This extends results from [32], [33]. Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky [7] by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov [40]. Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss in [8].