On the acyclic quantum cluster algebras with principal coefficients

In this paper, we focus on a new lower bound quantum cluster algebra which is generated by the initial quantum cluster variables and the quantum projective cluster variables of an acyclic quantum cluster algebra with principal coefficients. We show that the new lower bound quantum cluster algebra coincides with the corresponding acyclic quantum cluster algebra. Moreover, we establish a class of formulas between these generators and obtain the dual Poincar茅-Birkhoff-Witt (PBW) basis of this algebra.