Cohn Path Algebras of Higher-Rank Graphs

Abstract

Abstract. In this article, we introduce Cohn path algebras of higher-rank graphs. We prove that for a higher-rank graph Λ, there exists a higher-rank graph T Λ such that the Cohn path algebra of Λ is isomorphic to the Kumjian-Pask algebra of T Λ. We then use this isomorphism and properties of Kumjian-Pask algebras to study Cohn path algebras. This includes proving a uniqueness theorem for Cohn path algebras. IntroductionLeavitt path algebras were introduced and studied in [3] and [9] as a generalisation the class of algebras studied by Leavitt in [25]. Leavitt path algebras are also the natural algebraic analogues of graph C * -algebras studied in [32]; a number of interesting result have been proven by translating between graph C * -algebras and the ring theoretic Leavitt path algebras.Cohn path algebras were introduced in [6,7] and generalise the algebras U 1,n studied by Cohn in [16]. The idea is to build an algebra out of the path space of a graph; addition and scalar multiplication are defined formally and multiplication of two paths is only nonzero when one can concatenate the paths. Cohn path algebras can also be obtained from Leavitt path algebras by omitting one of the Cuntz-Krieger relations. Hence every Leavitt path algebra can be viewed as a quotient of a Cohn path algebra. On the other hand, Abrams, Ara and Siles Molina show in [2] that for every graph E, there exists a graph T E (denoted E (X) in [2]) such that the Cohn path algebra of E is isomorphic to the Leavitt path algebra of T E.Although it has received less attention, the Cohn path algebra of a directed graph is the algebraic analogue of the C * -algebraic Toeplitz algebra of E as defined in [19]. In the C * -algebraic setting, Muhly and Tomforde in [27] and also Sims in [38] each show that for a graph E, the Toeplitz algebra of E is isomorphic to the graph C * -algebra of T E. Now we move into the setting of higher-rank graph algebras: In [24], Kumjian and Pask introduced a combinatorial model, called a higher-rank graph, in order to capture the essential features of the C * -algebras studied by Robertson and Steger in [35]. A higherrank graph, also called a k-graph, is a generalisation of the path category of a directed graph where the length of a 'path' λ in a k-graph is an element of N k . Kumjian and Pask studied the C * -algebras associated to row-finite higher-rank graphs with no sources. Raeburn, Sims and Yeend generalised Kumjian and Pask's construction by describing the class of C * -algebras associated to more general higher-rank graphs in [30,31].1991 Mathematics Subject Classification. 16S99 (Primary); 16S10 (Secondary).

The Groupoid Approach to Leavitt Path Algebras

“…They are also very useful for studying structural properties like primeness and simplicity. Appropriate versions of these theorems have been proved not just for Leavitt path algebras but also (and we refer to [30,31,58,61]) for graph C * -algebras, as well as Cohn path algebras, higher-rank graph algebras, and even algebras of topological higher-rank graphs.…”

“…There are actually a few ways to associate a groupoid to a graph E; see for example [50, p. 511], [56, pp. 156-159], and [30,Example 5.4]. The one that we are interested in is called the boundary path groupoid, G E .…”