THE SOCLE AND SEMISIMPLICITY OF A KUMJIAN-PASK ALGEBRA

摘要

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. As a consequence of this structure theorem, every semisimple Kumjian-Pask algebra can be obtained as a Leavitt path algebra of a directed graph.