Leavitt path algebras of Cayley graphs $C_n^j$

TL;DR

This paper explicitly describes the Grothendieck groups of Leavitt path algebras associated with Cayley graphs of cyclic groups, using Smith Normal Form, and explores conditions for these groups to be infinite, with special focus on the case j=3.

Contribution

It provides a general, streamlined method to compute Grothendieck groups of Leavitt path algebras for Cayley graphs, extending previous specific cases.

Findings

Explicit formulas for Grothendieck groups using Smith Normal Form

Conditions for the groups to be infinite based on (j,n)

Connection between the case j=3 and Fibonacci-like sequences

Abstract

Let $n$ be a positive integer. For each $0\leq j \leq n-1$ 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 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.