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

Abstract

For an ample groupoid [Formula: see text] and a unit [Formula: see text] of [Formula: see text], Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra [Formula: see text] and the category of modules over the isotropy group algebra [Formula: see text]. In this paper, we prove a graded version of these functors and give a description of spectral graded simple modules over the graded Steinberg algebra [Formula: see text]. As an application, the spectral simple and graded simple modules over the Leavitt path algebra [Formula: see text] are classified. In particular, we show that many of previously known simple and graded simple [Formula: see text]-modules, including the Chen simple modules, are induced from (ungraded or graded) simple modules over isotropy group algebras.