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 Cqβ[Gr(k,n)], we
show that there is an associated braid group and each generator Οiβ of
the braid group preserves the quasi-commutative relations of quantum
Pl\"{u}cker coordinates and exchange relations of the quantum Grassmannian
cluster algebra. We conjecture that Οiβ also preserves r-term (rβ₯4) quantum Pl\"{u}cker relations of Cqβ[Gr(k,n)] and other
relations which cannot be derived from quantum quantum Pl\"{u}cker relations
(if any). Up to this conjecture, we show that Οiβ is a
quasi-automorphism of Cqβ[Gr(k,n)] and the braid group acts on
Cqβ[Gr(k,n)]