Leavitt path algebras of weighted Cayley graphs $C_n(S,w)$

TL;DR

This paper characterizes the Grothendieck groups of Leavitt path algebras derived from weighted Cayley graphs of cyclic groups, providing explicit descriptions and special case analyses.

Contribution

It offers a detailed description of the Grothendieck groups for Leavitt path algebras of weighted Cayley graphs, advancing understanding of their algebraic structure.

Findings

Explicit description of Grothendieck groups for these algebras

Analysis of Leavitt path algebras in specific cases

Enhanced understanding of algebraic invariants of weighted Cayley graphs

Abstract

For a postive integer $n$ and a subset $S$ of $\mathbb{Z}_n$, let $\left\langle S\right\rangle =\mathbb{Z}_n$, and $w:S\rightarrow\mathbb{N}$ be a function. The weighted Cayley graph of the cyclic group $\mathbb{Z}_n$ with respect to $S$ and $w$ is denoted by $C_n(S,w)$. We give an explicit description of the Grothendieck group of the Leavitt path algebras of $C_n(S,w)$. We also give description of Leavitt path algebras of $C_n(S,w)$ in some special cases.