On Cluster Algebras:Structure,Methods,Open Problems

We build and study the unfolding theory for sign-skew-symmetric cluster alge?bras, using unfolding theory, we solve some problems in cluster theory and prove some conjectures as well. Specially, we give a positive answer to an open problem asked by Berenstein-Fomin-Zelevinsky in 2005; we prove positivity conjecture, the conjectures that all cluster monomials are linear independent, F polynomials have constant 1 and the g-vectors are sign-coherent for acyclic sign-skew-symmetric clus-ter algebras. Meanwhile, we give the categorification for acyclic cluster algebras.Moreover, using triangular basis in quantum cluster algebras, we construct a series of bases for cluster algebras.This thesis is divided into three parts.(I) In the first part, we use the method of seed homomorphism to study the structure of cluster algebras. In particular, we classified all rooted cluster subalge-bras and a class of rooted cluster quotient algebras.(II) In the second part, we first do lots of preparations to prove the unfolding theorem for acyclic sign-skew-symmetric matrices. Secondly, we give the definition of unfolding for skew-symmetric matrices and the general construction of unfolding of acyclic sign-skew-symmetric matrices. Lately, we give the detail proof of the unfolding theorem for acyclic sign-skew-symmetric matrices.(?) The third part is the applications of the unfolding theorem of acyclic sign-skew-symmetric matrices. As a direct consequence, we give a positive answer for the problem asked by Berenstein-Fomin-Zelevinsky: any acyclic sign-skew-symmetric matrix is always totally sign-skew-symmetric. Hence, any acyclic sign-skew-symmetric matrix can be an exchange matrix for cluster algebras. In the sequel, using unfold-ing theorem, we build a surjective algebra homomorphism from an (infinite rank)skew-symmetric cluster algebra to an acyclic cluster algebra, moreover, we show such algebra homomorphism keeps good combinatoric properties. By the surjective algebras homomorphism, we proved the positivity for acyclic cluster algebras and the F-polynomials have constant 1. Then, we build the cluster character for acyclic cluster algebras and prove the sign-coherence conjecture for g-vectors. Furthermore,using Qin and Berenstein-Zelevinsky triangular basis, we construct a basis of cluster algebras, specially, such basis containing all cluster monomials.