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Purely infinite simple Kumjian–Pask algebras

Authors
  • Larki, Hossein
Type
Published Article
Journal
Forum Mathematicum
Publisher
De Gruyter
Publication Date
Jul 21, 2017
Volume
30
Issue
1
Pages
253–268
Identifiers
DOI: 10.1515/forum-2017-0002
Source
De Gruyter
Keywords
License
Yellow

Abstract

Given any finitely aligned higher-rank graph Λ and any unital commutative ring R, the Kumjian–Pask algebra KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is known as the higher-rank generalization of Leavitt path algebras. After the characterization of simple Kumjian–Pask algebras by Clark and Pangalela among others, in this article we focus on the purely infinite simple ones. Briefly, we show that if KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is simple and every vertex of Λ is reached from a generalized cycle with an entrance, then KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is purely infinite. We also prove a dichotomy for simple Kumjian–Pask algebras: If each vertex of Λ is reached only from finitely many vertices and KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is simple, then KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is either purely infinite or locally matritial. This result covers all unital simple Kumjian–Pask algebras.

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