Primitive algebraic algebras of polynomially bounded growth

TL;DR

This paper constructs finitely generated primitive algebraic algebras over countable fields with polynomially bounded growth, including a two-generated example, and discusses open problems in the area.

Contribution

It introduces new examples of primitive algebraic algebras with bounded Gelfand-Kirillov dimension, including a two-generated case, expanding understanding of algebraic growth constraints.

Findings

Existence of finitely generated primitive algebraic algebras with Gelfand-Kirillov dimension ≤ 6

Construction of a two-generated primitive algebraic algebra

Discussion of open problems in algebraic growth and structure

Abstract

We show that if $k$ is a countable field, then there exists a finitely generated, infinite-dimensional, primitive algebraic $k$-algebra $A$ whose Gelfand-Kirillov dimension is at most six. In addition to this we construct a two-generated primitive algebraic $k$-algebra. We also pose many open problems.