Quantum cluster algebras

TL;DR

This paper introduces and studies quantum deformations of cluster algebras, extending their algebraic framework to quantum analogs, which are relevant for total positivity and canonical bases in semisimple groups.

Contribution

It presents the first systematic study of quantum cluster algebras, expanding the classical theory into the quantum domain.

Findings

Defined quantum cluster algebras and their properties

Established connections with quantum groups and total positivity

Provided foundational results for further research in quantum algebra

Abstract

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in math.RT/0104151; their study continued in math.RA/0208229, math.RT/0305434. This is a family of commutative rings designed to serve as an algebraic framework for the theory of total positivity and canonical bases in semisimple groups and their quantum analogs. In this paper we introduce and study quantum deformations of cluster algebras.