The socle and semisimplicity of a Kumjian-Pask algebra

Abstract

The Kumjian-Pask algebra KP(\Lambda) is a graded algebra associated to a higher-rank graph \Lambda and is a generalization of the Leavitt path algebra of a directed graph. We analyze the minimal left-ideals of KP(\Lambda), and identify its socle as a graded ideal by describing its generators in terms of a subset of vertices of the graph. We characterize when KP(\Lambda) is semisimple, and obtain a complete structure theorem for a semisimple Kumjian-Pask algebra.