Leavitt Path Algebras of Cayley Graphs $$C_n^j$$ C n j

Abstract

Let n be a positive integer. For each $$0\le j \le n-1$$ we let $$C_n^j$$ denote the Cayley graph of the cyclic group $${\mathbb {Z}}_n$$ with respect to the subset $$\{1,j\}$$ . Utilizing the Smith Normal Form process, we give an explicit description of the Grothendieck group of each of the Leavitt path algebras $$L_K(C_n^j)$$ for any field K. Our general method significantly streamlines the approach that was used in a previous work to establish this description in the specific case $$j=2$$ . Along the way, we give necessary and sufficient conditions on the pairs (j, n) which yield that this group is infinite. We subsequently focus on the case $$j = 3$$ , where the structure of this group turns out to be related to a Fibonacci-like sequence, called the Narayana’s cows sequence.