The multiplication theorem and bases in finite and affine quantum cluster algebras

TL;DR

This paper proves a multiplication theorem for quantum cluster algebras of acyclic quivers, generalizing existing formulas, and constructs integral bases for finite and affine types that specialize to known bases.

Contribution

It introduces a generalized multiplication theorem for quantum cluster algebras and constructs new integral bases for finite and affine types.

Findings

Established a multiplication formula for quantum cluster variables.

Constructed $\\mathbb{ZP}$-bases in quantum cluster algebras.

Bases specialize to known integral bases when setting $q$ and coefficients to 1.

Abstract

We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in \cite{fanqin}. We apply the formula to construct some $\mathbb{ZP}$-bases in quantum cluster algebras of finite and affine types. Under the specialization $q$ and coefficients to $1$, these bases are the integral bases of cluster algebra of finite and affine types (see \cite{CK1} and \cite{DXX}).