Leavitt path algebras having Graded Invariant Basis Number

TL;DR

This paper characterizes when Leavitt path algebras of finite graphs possess the Graded Invariant Basis Number property, using matrix theory and graph classes, and explores its preservation under certain algebraic operations.

Contribution

It provides a complete matrix-theoretic characterization of gr-IBN failure in Leavitt path algebras of finite graphs and identifies classes where gr-IBN holds.

Findings

Leavitt path algebras of graphs with sinks have gr-IBN.

Cayley and Hopf graphs' Leavitt path algebras have gr-IBN.

gr-IBN is preserved under quotients by hereditary saturated subsets and Cartesian products.

Abstract

In this paper, we study the Graded Invariant Basis Number (grIBN) property for Leavitt path algebras of finite graphs. Using the talented monoid as our main tool, we establish a complete matrix-theoretic characterization of when a Leavitt path algebra of a finite graph fails to have gr-IBN. Consequently, we identify several classes of graphs whose Leavitt path algebras have gr-IBN, including graphs with sinks, Cayley graphs, and Hopf graphs associated with finite groups. We also investigate the preservation of gr-IBN under quotients by hereditary saturated subsets and under Cartesian products of graphs.