Topological Study Of Cluster Quivers And Pre-modulations With Potential

The main results of this paper are divided into four parts.First, we discuss the genuses of cluster quivers of finite mutation type. Almost all cluster quivers of finite mutation type come from triangulation of surface except for11exceptional quivers. We first give the table of genus distribution of11exceptional quivers by using Keller’s software quiver mutation in Java. We find that:the mutation equivalent classes of X6and X7contain cluster quivers of genus1while all quivers in other mutation equivalent classes are planar. For the relationship between the genus of an oriented surface and the genus of a cluster quiver from this surface, we give a partial answer. The genus of cluster quiver from surface will always be less than or equal to the genus of the surface. Moreover, we prove that there always exists a triangulation of surface such that the genus of its corresponding cluster quiver is equal to that of the surface whenever there are a certain number of punctures by using the tools from topological graph theory and algebraic topology. In the end of this part, we give a relation between the genus of a block-decomposable quiver and the number of block Ⅱ and block Ⅴ.The second part gives a more precise characterization of non-planar cluster quivers from surface. We first define skeleton graph of a non-planar graph and strongly block-decomposable graph only consisting of block Ⅰ and block Ⅱ. As the notions of smooth and subdivision in topological graph theory, we introduce admissible smooth and ad-missible subdivision both of which preserve strong block-decomposability. We prove that all skeleton graphs in non-planar block-decomposable graphs are strongly block-decomposable and give some characterization of principal paths in skeleton graphs. For the classification of reduced skeleton graphs of K5and K3.3type, we introduce useful notions such as basic component family, neighbored symbol type, symbol quiver and so on. Group action on set plays an important role in the proof of these classification. Classification Theorem asserts that:there are only7reduced skeleton graphs of K5type and16reduced skeleton graphs of K3.3type up to skeleton graph isomorphism. As an easy result of Classification Theorem, we give an analogue of Kuratowski’s theorem for non-planar cluster quivers from surface.In the third part, we generalize Derkeson-Weyman-Zelevinsky’s work on quivers with potential to the skew-symmetrizable case. Our main tool is pre-modulation, and we request all algebras on vertices should be semi-simple. We first define the partial derivative on basis element and then introduce Jacobian ideal and Jacobian algebra. As in [13], pre-modulations with potential also have Splitting Theorem, that is, can split into reduced part and trivial part. We define the mutation of pre-modulation with po-tential. At the end of this part, the decorated representations and their mutations are defined.Finally, as an earlier work of this research, we give a generalized Wedderburn prin-cipal theorem for complete algebras in I-adic topology and moreover give some charac-terizations for certain infinite dimensional algebras. The original motivation of this part is to give some characterizations for complete path algebras defined in the third part. Since this part is relatively independent, it is put in the last part.