Leavitt path algebras satisfying a polynomial identity

TL;DR

This paper characterizes Leavitt path algebras satisfying polynomial identities in terms of graph structure and algebraic properties, revealing connections between graph cycles, Gelfand-Kirillov dimension, and algebraic constraints.

Contribution

It provides a complete characterization of Leavitt path algebras satisfying polynomial identities using graph-theoretic and algebraic criteria, including dimension and cycle conditions.

Findings

L satisfying a polynomial identity is characterized by graph and algebraic conditions.

For finite graphs, polynomial identity satisfaction is equivalent to Gelfand-Kirillov dimension ≤ 1.

L has Gelfand-Kirillov dimension zero iff E has no cycles.

Abstract

Leavitt path algebras L of an arbitrary graph E over a field K satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When E is a finite graph, L satisfying a polynomial identity is shown to be equivalent to the Gelfand-Kirillov dimension of L being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph E, the Leavitt path algebra L has Gelfand-Kirillov dimension zero if and only if E has no cycles. Likewise, L has Gelfand-Kirillov dimension one if and only if E contains at least one cycle, but no cycle in E has an exit.