The Leavitt Path Algebras of Generalized Cayley Graphs

Abstract

Let n be a positive integer. For each 0 ≤ j ≤ n − 1 we let C j n denote Cayley graph for the cyclic group Zn with respect to the subset {1, j}. For any such pair (n, j) we compute the size of the Grothendieck group of the Leavitt path algebra L K (C j n ); the analysis is related to a collection of integer sequences described by Haselgrove in the 1940's. When j = 0, 1, or 2, we are able to extract enough additional information about the structure of these Grothendieck groups so that we may apply a Kirchberg-Phillips-type result to explicitly realize the algebras L K (C j n ) as the Leavitt path algebras of graphs having at most three vertices. The analysis in the j = 2 case leads us to some perhaps surprising and apparently nontrivial connections to the classical Fibonacci sequence.

K-theory for Leavitt path algebras: Computation and classification

“…Furthermore, if E is also a finite graph, then K top 0 (C * (E)) is a finitely generated abelian group and K top 1 (C * (E)) is isomorphic to the free part of K top 0 (C * (E)), so that C * (E) is classified up to Morita equivalence by the single group K top 0 (C * (E)). On the algebraic side, as shown in [2], [5], and [3] one may perform a similar construction to produce algebras from graphs. If E is a graph and k is any field, then one may mimic the graph C * -algebra construction to produce a k-algebra L k (E), which is called the Leavitt path algebra of E over k. For a given graph E, the algebra L k (E) (for any field k) has many properties in common with C * (E).…”

Subsets of Vertices Give Morita Equivalences of Leavitt Path Algebras

“…A finite path If E has a head, we can get a new graph F by collapsing the head down to a source. This is an example of a desingularisation and hence L R (F) and L R (E) are Morita equivalent by [1,Proposition 5.2]. Thus, the 'no-heads' hypothesis in Theorem 4.1 below is not restrictive.…”

Cycles in Leavitt path algebras by means of idempotents

“…The following breakthrough was to remove the hypothesis of row-finiteness in the underlying graphs. This was first done for Leavitt path algebras in [4] and [30]. It is often the case that the row-finite results are no longer valid for not necessarily row-finite graphs, and when they are, they may come up from totally different proofs, because the existence of infinite emitters (vertices that emit an infinite number of edges) disrupts the application of the (CK2) condition, fundamental to many established proofs, and therefore it causes new phenomena and forces the necessity of finding new tools to circumvent either the application of (CK2) or the appearance of infinite emitters.…”

“…It is often the case that the row-finite results are no longer valid for not necessarily row-finite graphs, and when they are, they may come up from totally different proofs, because the existence of infinite emitters (vertices that emit an infinite number of edges) disrupts the application of the (CK2) condition, fundamental to many established proofs, and therefore it causes new phenomena and forces the necessity of finding new tools to circumvent either the application of (CK2) or the appearance of infinite emitters. For example, in [4] it is shown that from a row-infinite countable graph E one can construct a row-finite countable graph F , a desingularization of E, in such a way that L K (E) and L K (F ) are Morita equivalent and there is a monomorphism of K-algebras from L K (E) to L K (F ). Finally and very recently, Leavitt path algebras have entered their final stage in terms of cardinality restrictions: by dropping also the countability assumption, arbitrary graphs are now the subject of study, a road started in [22] and [8].…”