Quasi-homomorphisms of quantum cluster algebras

In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasi-homomorphisms of cluster algebras introduced by Fraser. For a quantum Grassmannian cluster algebra ${\mathbb{C}}_q[\mathrm{G}\mathrm{r}(k,n)]$, we show that there is an associated braid group and each generator ${\sigma }_i$ of the braid group preserves the quasi-commutative relations of quantum Plücker coordinates and exchange relations of the quantum Grassmannian cluster algebra. We conjecture that ${\sigma }_i$ also preserves $r$-term ($r\ge 4$) quantum Plücker relations of ${\mathbb{C}}_q[\mathrm{G}\mathrm{r}(k,n)]$ and other relations which cannot be derived from quantum quantum Plücker relations (if any). Up to this conjecture, we show that ${\sigma }_i$ is a quasi-automorphism of ${\mathbb{C}}_q[\mathrm{G}\mathrm{r}(k,n)]$ and the braid group acts on ${\mathbb{C}}_q[\mathrm{G}\mathrm{r}(k,n)]$.