Generalized quantum cluster algebras: the Laurent phenomenon and upper bounds

TL;DR

This paper proves the Laurent phenomenon for generalized quantum cluster algebras and shows the equivalence of upper bounds and upper cluster algebras under certain conditions, advancing understanding of their algebraic structure.

Contribution

It establishes the Laurent phenomenon and the equality of upper bounds and upper cluster algebras for generalized quantum cluster algebras, extending known results to a broader class.

Findings

Laurent phenomenon holds in generalized quantum cluster algebras

Upper bounds coincide with upper cluster algebras under coprimality

Advances structural understanding of generalized quantum cluster algebras

Abstract

Generalized quantum cluster algebras introduced in [1] are quantum deformation of generalized cluster algebras of geometric types. In this paper, we prove that the Laurent phenomenon holds in these generalized quantum cluster algebras. We also show that upper bounds coincide with the corresponding generalized quantum upper cluster algebras under the "coprimality" condition.