Leavitt path algebras having Unbounded Generating Number

TL;DR

This paper characterizes when Leavitt path algebras have the Unbounded Generating Number property, linking it to graph conditions, and explores related algebraic properties like finiteness and projective cancellation.

Contribution

It provides necessary and sufficient graph conditions for Leavitt path algebras to possess the UGN property and related algebraic features, extending previous invariance results.

Findings

Leavitt path algebra has UGN iff the underlying graph satisfies specific conditions.

Identifies graphs for which the algebra is directly finite, stably finite, Hermite, and has projective cancellation.

Establishes UGN as a Morita invariant for unital rings.

Abstract

We present a result of P. Ara which establishes that the Unbounded Generating Number property is a Morita invariant for unital rings. Using this, we give necessary and sufficient conditions on a graph $E$ so that the Leavitt path algebra associated to $E$ has UGN. We conclude by identifying the graphs for which the Leavitt path algebra is (equivalently) directly finite; stably finite; Hermite; and has cancellation of projectives.