A $\mathbb{Z}$-basis for the cluster algebra associated to an affine quiver

TL;DR

This paper constructs explicit integer bases for cluster algebras of affine types and provides an inductive method for multiplying generalized cluster variables within tubes.

Contribution

It introduces $Z$-bases for affine type cluster algebras and an inductive formula for their multiplication, extending finite and rank 2 cases.

Findings

Explicit $Z$-bases for affine types $ ilde{A}_{n,n}$, $ ilde{D}$, $ ilde{E}$

Inductive formula for multiplication of generalized cluster variables

Extension of known bases from finite and rank 2 cases

Abstract

The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in \cite{CK2005} and \cite{SZ} respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras for affine types $\widetilde{A}_{n,n},\widetilde{D}$ and $\widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.