Atomic bases of quantum cluster algebras of type [formula omitted

Abstract

Let Q be the affine quiver of type A˜2n−1,1 and Aq(Q) be the quantum cluster algebra associated to the valued quiver (Q,(2,2,…,2)). We prove some cluster multiplication formulas, and deduce that the cluster variables associated with vertices of Q satisfy a quantum analogue of the constant coefficient linear relations. We then construct two bar-invariant Z[q±12]-bases B and S of Aq(Q) consisting of positive elements, and prove that B is an atomic basis.