Iterative algebras

TL;DR

This paper introduces iterative algebras constructed from morphisms on free monoids, characterizes their properties using linear algebra and combinatorics, and applies this to answer open questions in algebra.

Contribution

It provides a natural construction of iterative algebras, characterizes their properties, and demonstrates decidability, also solving specific open problems in algebra.

Findings

Many properties are decidable via linear algebra and combinatorics.

Constructed a primitive graded nilpotent algebra with Gelfand-Kirillov dimension two.

Answered open questions about finitely generated Lie algebras.

Abstract

Given a finitely generated free monoid $X$ and a morphism $\phi : X\to X$, we show that one can construct an algebra, which we call an iterative algebra, in a natural way. We show that many ring theoretic properties of iterative algebras can be easily characterized in terms of linear algebra and combinatorial data from the morphism and that, moreover, it is decidable whether or not an iterative algebra has these properties. Finally, we use our construction to answer several questions of Greenfeld, Leroy, Smoktunowicz, and Ziembowski by constructing a primitive graded nilpotent algebra with Gelfand-Kirillov dimension two that is finitely generated as a Lie algebra.