Integral quantum cluster structures

Abstract

We prove a general theorem for constructing integral quantum cluster algebras over $\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$, 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 $\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$. 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 ${\mathit{A}}_{\mathit{q}}{({\mathfrak{n}}_{+}(\mathit{w}))}_{\mathbb{Z}[{\mathit{q}}^{\pm 1}]}$ of the corresponding quantum unipotent cell has the property that ${\mathit{A}}_{\mathit{q}}{({\mathfrak{n}}_{+}(\mathit{w}))}_{\mathbb{Z}[{\mathit{q}}^{\pm 1}]}{\otimes }_{\mathbb{Z}[{\mathit{q}}^{\pm 1}]}\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$ is isomorphic to a quantum cluster algebra over $\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$ and to the corresponding upper quantum cluster algebra over $\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$.