1608.07744Aug 27, 2016V2: The characterization of finite dimensional Kumjian-Pask algebras is removed due to the referee's suggestion. Some changes are made and some typos are corrected
Given any finitely aligned higher-rank graph Λ and any unital commutative ring R, the Kumjian-Pask algebra KPR(Λ) is known as the higher-rank generalization of Leavitt path algebras. After characterizing simple Kumjian-Pask algebras by L.O. Clark and Y.E.P. Pangalela (and others), we focus in this article on the purely infinite simple ones. Briefly, we show that if KPR(Λ)
Nearby in the stack
is simple and every vertex of
Λ
is reached from a generalized cycle with an entrance, then
KPR(Λ)
is purely infinite. We next prove a standard dichotomy for simple Kumjian-Pask algebras: in the case that each vertex of
Λ
is reached only from finitely many vertices and
KPR(Λ)
is simple, then
KPR(Λ)
is either purely infinite or locally matritial. This result covers all unital simple Kumjian-Pask algebras.
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