Leavitt path algebras with bounded index of nilpotence

TL;DR

This paper characterizes Leavitt path algebras with bounded index of nilpotence through graph conditions, showing they satisfy polynomial identities and possess specific algebraic properties, thus answering an open question in the field.

Contribution

It provides a complete graphical characterization of Leavitt path algebras with bounded nilpotence index and links this property to polynomial identities and algebraic finiteness conditions.

Findings

Leavitt path algebra has bounded nilpotence index iff no cycle has an exit and path counts are bounded.

Such algebras satisfy polynomial identities.

They are directly-finite and $bZ$-graded $bSigma$-$V$ rings.

Abstract

In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence. We show that the Leavitt path algebra $L_{K}(E)$ has index of nilpotence at most $n$ if and only if no cycle in the graph $E$ has an exit and there is a fixed positive integer $n$ such that the number of distinct paths that end at any given vertex $v$ (including $v$, but not including the entire cycle $c$ in case $v$ lies on $c$) is less than or equal to $n$. Interestingly, the Leavitt path algebras having bounded index of nilpotence turn out to be precisely those that satisfy a polynomial identity. Furthermore, Leavitt path algebras with bounded index of nilpotence are shown to be directly-finite and to be $\mathbb{Z}$-graded $\Sigma$-$V$ rings. As an application of our results, we answer an open question raised in \cite{JST} whether an exchange $\Sigma$-$V$ ring has bounded index of nilpotence.