From quantum groups to quantum cluster algebras
Changjian Fu, Haicheng Zhang
TL;DR
This work builds a concrete bridge between quantum groups and quantum cluster algebras by constructing a homomorphism from the positive part of the quantum group $U_v^+(g)$ to a quantum cluster algebra $A_q$ with principal coefficients, using a Hall-algebra and Caldero–Chapoton framework. It develops twisted and standard compatible pairs and shows that quantum cluster variables arising from one-step mutations satisfy high-order fundamental and Serre relations within the quantum cluster algebra. By identifying appropriate seeds and maps between Hall algebras and quantum tori, the authors derive quantum Serre relations in $A_q(Q)$ and extend these to higher-order identities, generalizing and unifying representation-theoretic and cluster-algebraic approaches. The results provide a canonical realization of quantum Serre relations inside quantum cluster algebras and illuminate the structural links between rings of representations, Hall algebras, and cluster algebras with principal coefficients. The constructions are supported by explicit algebraic maps and an illustrative example, highlighting the practical impact for understanding quantum symmetries via cluster algebras.
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 $\mathcal {A}_q$.