Cluster algebra structure on the finite dimensional representations of affine quantum group Uq（A3）

In this paper, we prove one case of conjecture given by Hemandez and Leclerc. We give a cluster algebra structuure on the Grothendieck ring of a full subcategory of the finite dimensional representations of affine quantum group Uq（A3）. As a conclusion, for every exchange relation of cluster algebra, there exists an exact sequence of the full subcategory corresponding to it.