Integral cluster structures on quantized coordinate rings

TL;DR

This paper establishes quantum cluster algebra structures on the coordinate rings of complex simple algebraic groups over various rings, extending previous results and providing new integral and classical versions.

Contribution

It develops quantum cluster algebra structures over arbitrary rings for coordinate rings of simple algebraic groups, including integral forms and classical analogs, expanding the scope of cluster algebra applications.

Findings

Integral form of quantized coordinate ring admits an upper quantum cluster algebra structure.

Quantized coordinate rings of certain groups admit quantum cluster algebra structures over specified rings.

Classical versions of these structures hold over rings like ' and the integral form of type F4 is an upper cluster algebra.

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 $\Bbbk$.