| This paper mainly prove the conjecture given by Hernandez and Leclerc for the case of g of type A3, ? = 2[45]. Specially, there exists a full subcategory C2 of ?nite dimensional representations of quantum a?ne algebra Uq( A3) such that C2 is a monoidal categori?cation of cluster algebra A( Γ2). That is to say, the Grothendieck ring R2 of C2is isomorphic to A( Γ2), the cluster monomials of A( Γ2) are the classes of all the real simple objects of C2 and the cluster variables of A( Γ2)(including the frozen ones) are the classes of all the real prime simple objects of C2.In chapter 2, we introduce some related de?nitions and conclusions, including the theory of quantum a?ne algebras and cluster algebras.In chapter 3, we give a equivalence between theorem 3.1 and proposition 3.1. And then we prove that cluster algebra A( Γ) is isomorphic to the Grothendieck ring R2 of C2.The cluster algebra A( Γ2) de?ned in theorem 3.1 is of type E6, but the principal part of initial quiver Γ2is not an orientation of Dynkin graph E6. In order to using the theory of compatibility of roots, we choose a bipartite quiver Γ as initial quiver. In fact, the Grothendieck ring R2 of C2is a polynomial ring, and by the tool of T-systems, we ?nd the generators of cluster algebra A( Γ) such that A( Γ) is isomorphic to R2.In chapter 4, we calculate the truncated q-characters of all real prime simple objects of C2. Since q-characters are determined by its dominant monomials, for any object V of C, we de?ne truncated q-characters χq(V)≤4 which contain all dominant monomials.For the simple object L(m) with highest weight monomial m, we de?ne a polynomial N(m) ≤ χq(L(m)) and similarly we de?ne N(m)≤4. After calculating, it is pointed that for the real prime simple objects of C2, we have χq(L(m))≤4 = N(m)≤4.In chapter 5, we ?nish the proof of proposition 3.1. The cluster monomials are the classes of all the real simple objects; the cluster variables(including the frozen ones)are the classes of all the real prime simple objects. In order to describe the cluster monomials, we have an accurate description of compatible subsets of E6. As a conclusion,for every exchange relation of cluster algebra A( Γ), there exists an exact sequence of C2 corresponding to it. The set of such exact sequences includes the T-systems and coincides with extended T-systems de?ned in [63]. |
