Integral cluster structures on quantized coordinate rings
Hironori Oya, Fan Qin, Milen Yakimov
TL;DR
This work constructs and analyzes (quantum) cluster algebra structures on both quantized and classical coordinate rings of connected simply-connected complex simple groups. Leveraging Berenstein–Zelevinsky seeds, canonical bases, and triangular bases, it proves that integral forms $R_q[G]_{\mathbb{A}}$ (and their specializations) are upper cluster algebras and identifies when they coincide with (quantum) cluster algebras across Lie types, with particular attention to exceptional types like $E_8$ and $G_2$. A key result is that generalized quantum minors generate the coordinate rings over appropriate base rings, and matrix coefficients of the quantum adjoint representation provide explicit, representation-theoretic generators, enabling a uniform cluster-algebraic description across quantum and classical settings. The paper also shows stability under specialization to arbitrary commutative rings $\Bbbk$, via common triangular bases, enabling broad applicability of the cluster structure to algebraic and geometric contexts. Overall, it unifies quantum and classical coordinate rings within the cluster-algebra framework and clarifies generation, specialization, and seed-based realizations for a wide range of Lie types.
Abstract
We develop (quantum) cluster algebra structures over arbitrary commutative unital rings $\Bbbk$ and prove that the (quantized) coordinate rings of connected simply-connected complex simple algebraic groups $G$ over $\Bbbk$ admit such structures. We first show that the integral form of the quantized coordinate ring of $G$ admits an upper quantum cluster algebra structure over $\mathbb{A}=\mathbb{Z}[q^{\pm\frac{1}{2}}]$ by using a combination of tools from quantum groups, canonical bases and cluster algebras and a previous result of the second and third authors over $\mathbb{Q}(q^{\frac{1}{2}})$. We then obtain (integral) quantum versions of recent results of the first author: when $G$ is not of type $F_4$, the quantized coordinate ring of $G$ admits a quantum cluster algebra structure over $\mathbb{A}'$, where $\mathbb{A}'=\mathbb{A}$ when $G$ is not of types $G_2$, $E_8$, and $F_4$; $\mathbb{A}'=\mathbb{A}[(q^2+1)^{-1}]$ when $G$ is of type $G_2$, and $\mathbb{A}'=\mathbb{Q}(q^{\frac{1}{2}})$ when $G$ is of type $E_8$. We furthermore prove that the classical versions of these results hold over $\mathbb{A}'$ (where $\mathbb{A}'=\mathbb{Z}$ if $G$ is not of type $F_4$ or $G_2$ and $\mathbb{A}'=\mathbb{Z}[\frac{1}{2}]$ if $G$ is of type $G_2$) and that the integral form of the coordinate ring of $G$ of type $F_4$ is an upper cluster algebra. Finally, by using common triangular bases of (quantum) cluster algebras, we prove that the above results also hold under specializations of $\mathbb{A}$ and $\mathbb{A}'$ to commutative unital rings $\Bbbk$.