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In this paper, we prove one case of the conjecture given by Hernandez and Leclerc 20 .Specifically, we give a cluster algebra structure on the Grothendieck ring of a full subcategory of the finite dimensional representations of a simply-laced quantum affine algebra U q ( g). In the procedure, we also give a specific description of compatible subsets of type E 6 . As a conclusion, for every exchange relation of cluster algebra there exists a exact sequence of the full subcategory corresponding to it.

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Monogamy relations characterize the distributions of quantum entanglement in multipartite systems. In this work, we present some tighter monogamy relations in terms of the power of the Tsallis-q and Rényi-α entanglement in multipartite systems. We show that these new monogamy relations of multipartite entanglement with tighter lower bounds than the existing ones. Furthermore, three examples are given to illustrate the tightness.

In [8], the authors get a new presentation of two-parameter quantum algebra U v,t (g). Their presentation can cover all Kac-Moody cases. In this paper, we construct a suitable Hopf pairing such that U v,t (sl n ) can be realized as Drinfeld double of certain Hopf subalgebras with respect to the Hopf pairing. Using Hopf pairing, we construct a R-matrix for U v,t (sl n ) which will be used to give the Schur-Weyl dual between U v,t (sl n ) and Hecke algebra H k (v, t). Furthermore, using the Fusion procedure we construct the primitive orthogonal idempotents of H k (v, t). As a corollary, we give the explicit construction of irreducible U v,t (sl n )-representations of V ⊗k .