Kumjian-Pask algebras of higher-rank graphs. (Q2846977)

The authors introduce higher-rank analogues of Leavitt path algebras, called Kumjian-Pask algebras. To describe the construction one must start from a graph of rank \(k\) (or \(k\)-graph). This is defined as a category \(\Gamma\) together with a functor \(d\colon\Gamma\to\mathbb N^k\) satisfying a certain factorization property. Next the notion of Kumjian-Pask \(\Gamma\)-family is given. Roughly speaking, any such family \((P,S)\) is related to a family of orthogonal idempotents \(P_v\) associated to vertices, and to a family of tripotents \(S_\lambda\) related by identities that resemble those of Leavitt path algebras but are substantially more complex. Then for any row-finite \(k\)-graph without sources and any commutative ring \(R\) with \(1\) there is a Kumjian-Pask \(\Gamma\)-family \((p,s)\) and an \(R\)-algebra \(KP_R(\Gamma)\) satisfying a suitable universal property. This algebra is unique up to isomorphism and it is called the Kumjian-Pask algebra of \(\Gamma\) and the universal Kumjian-Pask \(\Gamma\)-family \((p,s)\). Leavitt path algebras arise as particular cases of this construction. In fact the class of Leavitt path algebras is strictly contained in the class of Kumjian-Pask ones, as the authors show in Remark 7.2. The next step is to prove that these algebras are \(\mathbb Z_k\)-graded (in the Leavitt path algebra case \(k=1\) and we get the canonical \(\mathbb Z\)-grading).NEWLINENEWLINE In section 4 they prove a graded-uniqueness theorem and a Cuntz-Krieger uniqueness theorem for \(KP_R(\Gamma)\). In the last sections the ideal structure of \(KP_R(\Gamma)\) is investigated. First, graded ideals have the usual description in terms of saturated hereditary subsets of vertices; then, the so called basic ideals are proved to be graded under suitable conditions. The graphs whose Kumjian-Pask algebra are basically simple are described. More generally, they prove that if \(\Gamma\) is a row-finite \(k\)-graph without sources then \(KP_R(\Gamma)\) is simple if and only if \(R\) is a field and \(\Gamma\) is aperiodic and cofinal.