Primitivity of finitely presented monomial algebras

TL;DR

This paper investigates the structure of prime finitely presented monomial algebras, showing they are either primitive or have GK dimension one and satisfy a polynomial identity, extending to automaton algebras.

Contribution

It proves that prime finitely presented monomial algebras are either primitive or have GK dimension one and satisfy a polynomial identity, confirming a conjecture for automaton algebras.

Findings

Prime finitely presented monomial algebras are either primitive or have GK dimension one and satisfy a polynomial identity.

The result extends to automaton algebras, a broader class of monomial algebras.

This work confirms a conjecture by the first author and Smoktunowicz.

Abstract

We study prime monomial algebras. Our main result is that a prime finitely presented monomial algebra is either primitive or it has GK dimension one and satisfies a polynomial identity. More generally, we show this result holds for the class of \emph{automaton algebras}; that is, monomial algebras that have a basis consisting of the set of words recognized by some finite state automaton. This proves a special case of a conjecture of the first author and Agata Smoktunowicz.