Quasi-homomorphisms of quantum cluster algebras

In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasihomomorphisms 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 sigma i of the braid group preserves the quasi-commutative relations of quantum Plucker coordinates and exchange relations of the quantum Grassmannian cluster algebra. We conjecture that sigma i also preserves r -term (r >= 4) quantum Plucker relations of Cq[Gr(k, n)] and other relations which cannot be derived from quantum Plucker relations (if any). Up to this conjecture, we show that sigma i is a quasi-automorphism of Cq[Gr(k, n)] and the braid group acts on Cq[Gr(k, n)]. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).