From quantum groups to quantum cluster algebras

TL;DR

This paper establishes a homomorphism linking quantum groups to quantum cluster algebras and demonstrates that quantum cluster variables satisfy quantum Serre relations, revealing deep algebraic connections.

Contribution

It introduces a homomorphism from quantum groups to quantum cluster algebras and proves that cluster variables satisfy quantum Serre relations.

Findings

Established a homomorphism from quantum groups to quantum cluster algebras.

Proved quantum cluster variables satisfy quantum Serre relations.

Connected quantum group structures with cluster algebra mutations.

Abstract

We provide a homomorphism of algebras from the quantum group $\mathbf{U}^+_v(\mathfrak{g})$ to the corresponding quantum cluster algebra $\mathcal {A}_q$ with principal coefficients. As a by-product, we show that the quantum cluster variables arising from one-step mutations from the initial cluster variables satisfy the (high order) quantum Serre relations in $\mathcal {A}_q$.