Properties of the gradings on ultragraph algebras via the underlying combinatorics

TL;DR

This paper characterizes the properties of two types of gradings on ultragraph Leavitt path algebras using the underlying combinatorial structure, revealing new insights especially for the free group grading.

Contribution

It provides a combinatorial characterization of when these gradings are strong or epsilon-strong, including new results for the free group grading in ultragraph and graph Leavitt path algebras.

Findings

Characterization of strong and epsilon-strong gradings in ultragraph Leavitt path algebras.

New results on free group gradings in the context of ultragraphs and graphs.

Relation between grading strongness and saturation of gauge actions in ultragraph C*-algebras.

Abstract

There are two established gradings on Leavitt path algebras associated with ultragraphs, namely the grading by the integers group and the grading by the free group on the edges. In this paper, we characterize properties of these gradings in terms of the underlying combinatorial properties of the ultragraphs. More precisely, we characterize when the gradings are strong or epsilon-strong. The results regarding the free group on the edges are new also in the context of Leavitt path algebras of graphs. Finally, we also describe the relation between the strongness of the integer grading on an ultragraph Leavitt path algebra and the saturation of the gauge action associated with the corresponding ultragraph C*-algebra.