On the Kurosh problem for algebras over a general field

TL;DR

This paper constructs new examples of infinite-dimensional finitely generated nil algebras over any field, with growth rates that are super-polynomial but subexponential, including the first such examples over uncountable fields.

Contribution

It extends the existence of nil algebras with controlled subexponential growth to uncountable fields, generalizing previous countable-field results.

Findings

Constructed nil algebras over any field with prescribed super-polynomial but subexponential growth.

First examples of nil algebras with subexponential growth over uncountable fields.

Demonstrated the existence of finitely generated nil algebras with specific growth bounds.

Abstract

Smoktunowicz, Lenagan, and the second-named author recently gave an example of a nil algebra of Gelfand-Kirillov dimension at most three. Their construction requires a countable base field, however. We show that for any field $k$ and any monotonically increasing function $f(n)$ which grows super-polynomially but subexponentially there exists an infinite-dimensional finitely generated nil $k$-algebra whose growth is asymptotically bounded by $f(n)$. This construction gives the first examples of nil algebras of subexponential growth over uncountable fields.