Purely infinite simple Kumjian–Pask algebras

Abstract

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.