On induced graded simple modules over graded Steinberg algebras with applications to Leavitt path algebras

TL;DR

This paper develops a graded version of induction and restriction functors for Steinberg algebras of graded ample groupoids and applies these to classify simple modules over Leavitt path algebras, including many known modules.

Contribution

It introduces graded induction and restriction functors for graded Steinberg algebras and applies them to classify simple modules over Leavitt path algebras, connecting modules to isotropy group algebras.

Findings

Classified spectral and graded simple modules over Leavitt path algebras.

Showed many known simple modules are induced from isotropy group algebra modules.

Extended Steinberg algebra module theory to a graded setting.

Abstract

For an ample groupoid $\mathcal{G}$ and a unit $x$ of $\mathcal{G}$, Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra $A_R(\mathcal{G})$ and the category of modules over the isotropy group algebra $R\mathcal{G}_x$. In this paper, we prove a graded version of these functors and related results for the graded Steinberg algebra of a graded ample groupoid. As an application, the spectral simple and graded simple modules over the Leavitt path algebra $L_K(E)$ are classified. In particular, we show that many of previously known simple and graded simple $L_K(E)$-modules, including the Chen simple modules, are induced from (graded or non-graded) simple modules over isotropy group algebras.