Integral cluster structures on quantized coordinate rings

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 $\Bbbk$.

💡 Research Summary

The paper develops a comprehensive framework for (quantum) cluster algebras over arbitrary commutative unital rings and applies it to the (quantized) coordinate rings of connected, simply‑connected complex simple algebraic groups. The authors first treat the integral form of the quantized coordinate ring (R_q