Quantum cluster algebras and their specializations

Abstract

We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be viewed as a flat deformation of the classical cluster algebra.

11 pages, v2: small corrections and improvements after referee report. Final version, to appear in J. Algebra

References in corpus (3)

• Kac-Moody groups and cluster algebras
• Cluster structures on quantum coordinate rings
• Graded quantum cluster algebras and an application to quantum Grassmannians

Cited by in corpus (6)

• Strong positivity for quantum theta bases of quantum cluster algebras
• Quantum Grothendieck rings as quantum cluster algebras
• $\mathcal{A}=\mathcal{U}$ for cluster algebras from moduli spaces of $G$-local systems
• Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems
• Root of unity quantum cluster algebras and discriminants
• Root of unity quantum cluster algebras and Cayley-Hamilton algebras