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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 principle 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 PBW basis of this algebra.
Huang et al. (Wed,) studied this question.
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