Integral quantum cluster structures

TL;DR

This paper proves a general theorem establishing integral quantum cluster algebra structures for quantum nilpotent algebras, extending previous results beyond acyclic cases, and applies it to quantum unipotent cells associated with Kac-Moody algebras.

Contribution

The paper introduces a broad theorem for integral quantum cluster structures on quantum nilpotent algebras, generalizing prior acyclic-only results and linking them to quantum unipotent cells.

Findings

Integral forms of quantum nilpotent algebras have quantum cluster algebra structures.

Quantum unipotent cells are isomorphic to quantum cluster algebras over ${f Z}[q^{rac{1}{2}}]$.

The results extend the scope of quantum cluster algebra theory to more general algebraic structures.

Abstract

We prove a general theorem for constructing integral quantum cluster algebras over ${\mathbb{Z}}[q^{\pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over ${\mathbb{Z}}[q^{\pm 1/2}]$. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that for every symmetrizable Kac-Moody algebra ${\mathfrak{g}}$ and Weyl group element $w$, the dual canonical form $A_q({\mathfrak{n}}_+(w))_{{\mathbb{Z}}[q^{\pm 1}]}$ of the corresponding quantum unipotent cell has the property that $A_q( {\mathfrak{n}}_+(w))_{{\mathbb{Z}}[q^{\pm 1}]} \otimes_{{\mathbb{Z}}[q^{ \pm 1}]} {\mathbb{Z}}[ q^{\pm 1/2}]$ is isomorphic to a quantum cluster algebra over ${\mathbb{Z}}[q^{\pm 1/2}]$ and to the corresponding upper quantum cluster algebra over ${\mathbb{Z}}[q^{\pm 1/2}]$.