Morphisms Between Cluster Algebras,cluster Structures And Cluster Automorphism Groups

The main objects of this thesis are morphisms between cluster algebras and cluster structures in 2-Calabi-Yau triangulated categories, in particular, the cluster automorphisms of a cluster algebra and the group of these automorphisms.We give the guidance and some preliminaries in the chapter one and the chapter two respectively. The third chapter is devoted to rooted cluster algebras and rooted cluster morphisms which were introduced in[1]recently. An example of rooted cluster morphism which is not ideal is given, this clarifies a doubt in[1]. We introduce the notion of frozenization of a seed and show that an injective rooted cluster morphism always arises from a frozenization and a subseed; moreover, it is a section if and only if it arises from a subseed. This answers the Problem 7.7 in[1]. We prove that an inducible rooted cluster morphism is ideal if and only if it can be decomposed as a surjective rooted cluster morphism and an injective rooted cluster morphism. We also introduce the tensor decompositions of a rooted cluster algebra and of a rooted cluster morphism.In the fourth chapter, we consider the relations between cluster algebras and cluster structures in 2-Calabi-Yau triangulated categories. For the rooted cluster algebras arising from a 2-Calabi-Yau triangulated category C with cluster tilting objects, we give an oneto-one correspondence between certain pairs of their rooted cluster subalgebras which we call complete pairs(see Definition 3.4 for precise meaning) and cotorsion pairs in C.In the fifth chapter, the cluster automorphism group of a cluster algebra with geometric coeffcients is studied systematically, where the automorphism group of a cluster algebra with trivial coeffcients has been studied in[2]. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra(i.e. the corresponding cluster algebra without coeffcients). We show that several classes of cluster algebras with coeffcients are gluing free, for example, cluster algebras with principal coeffcients, cluster algebras with universal geometric coeffcients, and cluster algebras from surfaces(except a 4-gon) with coeffcients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coeffcients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coeffcients, its cluster automorphism group is isomorphic to the automorphism group of its initial matrix. For a cluster algebra, we introduce the automorphism group of its exchange graph, and show that if the algebra is gluing free,then its cluster automorphism group is a subgroup of this group.The sixth chapter is devoted to a study of the automorphism group Aut(A) of a cluster algebra A of finite type. We show that except type D2 n,(n 2), by the wellknown correspondence between cluster variable set X and the corresponding almost positive root system Φ≥-1, all the cluster automorphisms of A are induced from two piecewise-linear transformations τ+and τ-[3]on Φ≥-1. For a cluster algebra of type D2 n,(n 2), there exists exceptional cluster automorphism induced by a permutation of negative simple roots in Φ≥-1which is not generated by τ+and τ-. By using these results and folding simple laced cluster algebras, we compute the cluster automorphism groups of non-simple laced finite type cluster algebras. We also show that Aut(A) is isomorphic to the automorphism group Aut(Auniv) for the FZ-universal cluster algebra Auniv [4]of A.Finally, in the seventh chapter, for a coeffcient free cluster algebra A, we study relations between the cluster automorphism group Aut(A) and the automorphism group Aut(EA) of its exchange graph EA. We show that these two groups are isomorphic with each other, if A is of finite type, excepting types of rank two and type F4, or A is of skew-symmetric finite mutation type.