Representations of relative Cohn path algebras

Abstract

We study relative Cohn path algebras, also known as Leavitt-Cohn path algebras, and we realize them as partial skew group rings (to do this we prove uniqueness theorems for relative Cohn path algebras). Furthermore, given any graph E we define E-relative branching systems and prove how they induce representations of the associated relative Cohn path algebra. We give necessary and sufficient conditions for faithfulness of the representations associated to E-relative branching systems (this improves previous results known to Leavitt path algebras of row-finite graphs with no sinks). To prove this last result we show first a version, for relative Cohn-path algebras, of the reduction theorem for Leavitt path algebras.

Irreducible and permutative representations of ultragraph Leavitt path algebras

“…Iterated function systems and branching systems are widely used in the study of representations of algebras associated to combinatorial objects, see for example [8,11,12,13,14,15,16,17,18,19,20,21,23,25,31]. Hence it is interesting to note that the representation π of Theorem 3.7 can be constructed via branching systems.…”

“…Remark 4.3 The above branching system can also be seen as a partial action of the free group on the edges of the ultragraph and be used to realized L R (G) as a partial skew group ring, see [8,22,24].…”

Representations and the reduction theorem for ultragraph Leavitt path algebras

“…The reduction theorem for Leavitt path algebras, see [1,7], is an extremely useful tool in estabilishing various ring-theoretic properties of Leavitt path algebras (for example, the uniqueness theorems for Leavitt path algebras follow with mild effort from the reduction theorem). A version for relative Cohn path algebras was given in [11], where it was also used as an important tool in the study of representations of these algebras. In our context the reduction theorem allow us to characterize faithful representations of Leavitt ultragraph path algebras arising from branching systems, but we expect it will also have applications in further studies of ultragraph Leavitt path algebras (for example, in Corollary 3.3 we show that ultragraph Leavitt path algebras are semiprime).…”

“…of Kumjian-Pask in [8], of Steinberg algebras in [3,10]. Representations of various algebras, in connection with branching systems, were studied in [11,15,17,18,20,21,22,23,24,25,27]. To describe the connections of representations of ultragraph Leavitt path algebras with branching systems is the second goal of this paper.…”