代数的导出等价理论及相关课题的研究

The research of derived categories of rings or algebras is very active in the representation theory. Derived equivalence is a basic relation between rings or algebras. Derived equivalence has been shown to preserve many invariants and provide new connection. Moreover, derived equivalence is related to the famous conjecture-Broue conjecture in the representation theory of groups, and many conjectures in the representation theory of finite dimensional algebras, such as finitistic dimension conjecture, Auslander conjecture. Today, they have widely been used in many branches: algebraic geometry, Lie algebra and mathematical physics, etc. Recently, we construct derived equivalences from known derived equivalent alebras, as well as the finiteness of the finitistic dimension is the invariant under derived equivlance, and our work has been attracted international attention. This topic will focus on derived equivalences and related equivalences and we will study the following problems: .(1) If the Auslander algebras of representation-finite selfinjective Artin algebras are derived equivalent, are the original algebras already derived equivalent ? .(2) Study the relationship between derived equivalences and cluster equivalences; .(3) Find the conditions from stable equivalences of Morita type to derived equivalences; .(4) Construct recollements from infinitely generated generalized tilting module; (5) Study the relationship between stable equivalence of Morita type and graded Lie algebra structure of Hochschild cohomology group.

导出等价是表示论研究中非常活跃的课题，它是环或代数之间一种基本的等价关系，与群表示论中的著名猜想Broue猜想以及代数表示论中许多著名的猜想,比如有限维数猜想,Auslander 猜想等想密切相关。它还在代数几何，李代数以及数学物理等领域有着重要的应用。最近，我们从已知代数的导出等价构造导出等价，以及有限维数的有限性是导出等价的不变量等工作，引起了国际同行的关注。本课题将围绕导出等价展开下列研究：.(1) 表示有限型自入射代数的Auslander代数的导出等价能否得到原来表示有限型自入射代数的导出等价；.(2) 导出等价与cluster 等价的关系；.(3) 从Morita 型稳定等价得到导出等价的条件；.(4) 从无限生成推广的倾斜模构造recollement；.(5) Morita 型稳定等价与Hochschild上同调群的分次李结构的关系。

导出等价是表示论研究中非常活跃的课题，它是环或代数之间一种基本的等价关系，与群表示论中的著名猜想Broue猜想以及代数表示论中许多著名的猜想,比如有限维数猜想,Auslander 猜想等想密切相关。它还在代数几何，李代数以及数学物理等领域有着重要的应用。本课题将围绕导出等价展开研究，取得了一系列成果。首先，证明了导出等价保持推广的Auslander-Reiten猜想。将导出等价的工具引入到Auslander-Reiten猜想的研究中，为解决该猜想提供了一条新的思路。其次，构造了 Φ-Cohen-Macaulay Auslander -Yoneda代数之间的导出等价。从代数的三角范畴中构造了与三角相关联的Φ-Green代数之间的导出等价。研究了标准的导出等价与Cohen-Macaulay Auslander代数之间的关系，将三角范畴中与Φ-Auslander-Yoneda代数相关的子代数之间的导出等价转化为自同态代数之间的导出等价。最后，证明了导出等价得到的稳定函子可以诱导Gorenstein投射模范畴之间的稳定等价。