The Leavitt Path Algebras of Generalized Cayley Graphs

Abstract

Let n be a positive integer. For each ${0 \leq j \leq n-1}$, we let ${C_{n}^{j}}$ denote Cayley graph for the cyclic group ${\mathbb{Z}_n}$ 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 ${L_K(C_{n}^{j})}$; the analysis is related to a collection of integer sequences described by Haselgrove in the 1940s. 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 ${L_K(C_{n}^{j})}$ 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.