The socle and semisimplicity of a Kumjian-Pask algebra
Jonathan H. Brown, Astrid an Huef
TL;DR
This paper investigates the structure of Kumjian-Pask algebras derived from higher-rank graphs, focusing on their socle, minimal left-ideals, and conditions for semisimplicity, providing a comprehensive structural characterization.
Contribution
It characterizes the socle and semisimplicity of Kumjian-Pask algebras, offering a complete structure theorem for semisimple cases, extending understanding beyond Leavitt path algebras.
Findings
Identified the socle as a graded ideal generated by specific vertices.
Provided necessary and sufficient conditions for semisimplicity.
Established a complete structure theorem for semisimple Kumjian-Pask algebras.
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.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Algebraic structures and combinatorial models · Advanced Topics in Algebra