Quasi-homomorphisms of quantum cluster algebras

Abstract

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 Cq[Gr(k,n)], we show that there is an associated braid group and each generator σi 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 σi also preserves r-term (r≥4) quantum Plücker relations of Cq[Gr(k,n)] and other relations which cannot be derived from quantum Plücker relations (if any). Up to this conjecture, we show that σi is a quasi-automorphism of Cq[Gr(k,n)] and the braid group acts on Cq[Gr(k,n)].