雅可比代数模范畴上的突变函子及其应用

We consider Jacobian module category of a quiver with potential (Q,W): the module category over a Jacobian algebra associated with (Q,W). We define and study left and right mutation functors between two Jacobian module categories which are induced by Derkson-Welman-Zelevinsky mutation of quivers with potentials. We show that these left and right mutation functors are a generalization of Bernstein-Gelfand-Ponomarev functors.. As an application on our mutation functors, we generalize a result of Ringel on cluster-concealed algebras, which is published in Adv. Math. to cluster-tilted algebras; Moreover, by using the link between cluster categories and Riemann sufaces, we are going to find the maximal length of so-called “maximal green sequences” induced by cluster-tilted algebra of type A, and we compare this length with the diameter problem in the Sleator-Tarjan-Thurston conjecture which is well-known in hyperbolic geometry and combinatorics theory.

本项目主要考虑一个带势箭图(Q,W)所对应雅可比代数J(Q,W)上的模范畴，简称为雅可比代数模范畴。项目主要定义、并研究由带势箭图(Q,W)的Derkson-Welman-Zelevinsky突变(Mutation)，诱导的两个雅可比代数模范畴之间的左右突变函子。项目目标之一是证明这个的雅可比代数模范畴上的突变函子，是经典的Bernstein-Gelfand-Ponomarev函子的推广。. 作为雅可比代数模范畴的突变函子的应用，项目将把Ringel发表在Adv. Math.上一篇关于Cluster-concealed代数的结论推广到一般的Cluster-tilted代数中去；项目还将利用丛范畴与黎曼曲面的联系，找出由A型丛代数诱导的极大绿色序列的长度的最大值，并将其与双曲几何、组合理论中著名的Sleator-Tarjan-Thurston猜想中的直径问题做比较。

本项目主要考虑一个带势箭图(Q,W)所对应雅可比代数J(Q,W)上的模范畴，简称为雅可比代数模范畴。项目主要定义、并研究由带势箭图(Q,W)的Derkson-Welman-Zelevinsky 突变(Mutation)，诱导的两个雅可比代数模范畴之间的左右突变函子。项目定义了这个的雅可比代数模范畴上的突变函子，并证明了这些突变函子是经典的Bernstein-Gelfand-Ponomarev函子的推广。即对于任意一个顶点诱导的两个突变函子都给出了两个子范畴的等价。项目还利用左右突变函子与黎曼曲面上的联系，考虑了由黎曼曲面诱导的一个相应的交换图。并证明了这个交换图满足“Non-leaving-face”性质。随后，我们又利用范畴表示理论与丛范畴理论的联系，证明了一般的2-Calabi-Yau三角范畴诱导的交换图也满足“Non-leaving-face”性质.