On the acyclic quantum cluster algebras with principal coefficients

TL;DR

This paper proves that a new lower bound quantum cluster algebra generated by specific variables coincides with the original acyclic quantum cluster algebra with principal coefficients, and derives formulas and the dual PBW basis.

Contribution

It introduces a new lower bound quantum cluster algebra and shows its equivalence to the original, also providing formulas and the dual PBW basis for the algebra.

Findings

The new lower bound quantum cluster algebra coincides with the acyclic quantum cluster algebra.

Formulas relating the generators of the algebra are established.

The dual PBW basis of the algebra is obtained.

Abstract

In this paper, we focus on a new lower bound quantum cluster algebra which is generated by the initial quantum cluster variables and the quantum projective cluster variables of an acyclic quantum cluster algebra with principal coefficients. We show that the new lower bound quantum cluster algebra coincides with the corresponding acyclic quantum cluster algebra. Moreover, we establish a class of formulas between these generators, and obtain the dual PBW basis of this algebra.