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

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.