导出范畴与稳定条件

The study of stability conditions on triangulated categories is a hot research topic in the representation theory of algebras, algebraic topology, geometry, and dynamic systems. It is closely related to the theory of moduli spaces, homological mirror symmetry, topological invariants (such as Donaldson-Thomas invariants and wall crossing formulas). As an important and special class of triangulated categories, the study of stability conditions on derived categories has attracted much attention.. This project mainly focuses on the topological properties and structures of the spaces of stability conditions on bounded derived categories of affine hereditary algebras, as well as of Calabi-Yau-N Ginzburg algebras of affine quivers. More precisely, we intend to describe the properties of bounded t-structures and exchange graphs of these two kinds of triangulated categories; to characterize the structures of connected components corresponding to exchange graphs in the spaces of stability conditions; and to prove the faithfulness of the action of the braid group and the simple connectedness of these spaces. This project will greatly promote the development of related fields.

三角范畴上的稳定条件是代数表示论、代数拓扑、几何和动力系统等数学分支交叉领域的一个热门研究课题，与模空间、同调镜面对称、拓扑不变量（如 Donaldson-Thomas 不变量与wall crossing公式）等理论都有紧密的联系。导出范畴作为一类重要且特殊的三角范畴，其上稳定条件的研究备受关注。. 本项目拟研究仿射遗传代数的有界导出范畴以及仿射箭图的Calabi-Yau-N Ginzburg代数的有限维导出范畴上的稳定条件空间的拓扑性质和结构。具体地，拟研究这两类三角范畴的有界t-结构以及Exchange图的性质；拟刻画稳定条件空间中与Exchange图对应的连通分支的结构；拟证明辫子群作用的忠实性以及稳定条件空间的单连通性。该项目将极大地推进相关研究领域的发展。

稳定条件是代数表示论、代数几何和量子群理论等数学分支交叉领域的一个热门研究课题，与模空间理论、Hall代数、几何不变量等理论都有紧密的联系。本项目主要探讨了一些特殊范畴上的稳定条件空间的拓扑性质和结构。具体地，我们通过凝聚层范畴系统地研究了三个顶点的仿射A型箭图的稳定条件空间的连通性，并通过稳定条件空间的刻画给出了有界t-结构的分类。我们定义了赋值加权射影线的凝聚层范畴，并证明了它与对应的赋值canonical代数是导出等价的。我们研究了量子群的有限维模的扭模，并证明了量子仿射A型代数的任何一个有限维不可约模和它的扭模都是同构的。我们探讨了阿贝尔长度范畴A的实Grothendieck群 K_0 (A) 的胞腔结构，证明了K_0 (A)的胞腔和内部非空的TF等价类是一一对应的，以及证明了K_0 (A)的墙与胞腔都是单连通的，并应用到了余模范畴。我们研究了路代数模范畴和导出范畴上的稳定条件的联系，并探讨了由稳定条件定义的HN系统和Ringel-Hall代数之间的关系。