Higher structures and deformations in representation theory

表示论中的高级结构和变形

• 批准号: 503982309
• 负责人: Dr. Zhengfang Wang
• 金额: --
• 依托单位: Nanjing University
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2022
• 资助国家: 德国
• 起止时间: 2021-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/503982309?language=en
• 关键词: Higher; structures; deformations; representation; theory

The research proposal is focusing on higher algebraic structures arsing from Hochschild, and related, cohomology and on higher categorical structures such as A-infinity categories. I aim to develop new and advanced techniques applicable to a wide range of problems in algebra, representation theory, topology, and geometry. Over the past few years, I have been investigating higher algebraic structures arising from singular Hochschild cohomology. I have shown that singular Hochschild cohomology is endowed with the same rich structure as classical Hochschild cohomology: a Gerstenhaber algebra in cohomology and a B-infinity algebra at the complex level. Suggested by this result, Keller proved that singular Hochschild cohomology is isomorphic to the Hochschild cohomology of the dg singularity category. He also conjectured that this isomorphism lifts to a quasi-isomorphism of B-infinity algebras. The first objective in my research proposal is to prove Keller's conjecture stated above. To achieve this, jointly with X. -W. Chen we introduced an explicit dg enhancement of singularities categories. In this proposal, we plan to construct a natural action of the B-infinity structure on singular Hochschild cohomology on this dg enhancement, which will yield a B-infinity quasi-isomorphism lifting Keller's isomorphism. The second objective is to explore further applications of singular Hochschild cohomology in topology and geometry. Jointly with M. Rivera we have shown that there is a rich higher algebraic structure on the singular Hochschild cohomology of any dg Frobenius algebra. In this proposal, we plan to show that this higher structure actually comes from a natural action of the Sullivan dg PROP, originally used to model operations in string topology. This proposal also concerns (A-infinity) deformations of (graded) algebras and their applications to representation theory. Recently jointly with S. Barmeier we have developed a combinatorial method to study the deformation theory of a path algebra of any finite quiver with relations, which allows us to easily compute particular classes of examples. The third objective aims to understand the behaviour of singularity categories under deformation quantisation, motivated by Keller's conjecture. Particularly, we expect an equivalence between singularity categories of a cyclic quotient singularity and of the deformation quantisation of a natural Poisson structure on the singularity. The fourth objective is to describe the A-infinity deformations of graded gentle algebras up to derived equivalence, in terms of their surface models. In particular, we expect that some A-infinity deformations correspond to doing surgery on the surface models.The fifth objective is to use our combinatorial method to study and verify Stroppel's conjecture for extended Khovanov arc algebras, which yields the intrinsic formality of these algebras.

该研究计划的重点是 Hochschild 提出的更高代数结构以及相关的上同调和更高的范畴结构，例如 A-无穷范畴。我的目标是开发适用于代数、表示论、拓扑和几何中广泛问题的新技术和先进技术。在过去的几年里，我一直在研究由奇异霍克希尔德上同调产生的高等代数结构。我已经证明奇异 Hochschild 上同调被赋予与经典 Hochschild 上同调相同的丰富结构：上同调中的 Gerstenhaber 代数和复级上的 B-无穷代数。根据这个结果，Keller 证明了奇异 Hochschild 上同调与 dg 奇点范畴的 Hochschild 上同构。他还推测这种同构提升为 B-无穷代数的准同构。我的研究计划的第一个目标是证明上述凯勒的猜想。为了实现这一目标，与 X.-W. 共同努力。 Chen 我们引入了奇点类别的显式 dg 增强。在这个提案中，我们计划在这个 dg 增强上构造奇异 Hochschild 上同调上的 B-无穷大结构的自然作用，这将产生提升 Keller 同构的 B-无穷大准同构。 第二个目标是探索奇异 Hochschild 上同调在拓扑和几何中的进一步应用。 我们与 M. Rivera 一起证明了任何 dg Frobenius 代数的奇异 Hochschild 上同调上都存在丰富的高等代数结构。在这个提案中，我们计划证明这种更高的结构实际上来自 Sullivan dg PROP 的自然作用，最初用于对弦拓扑中的操作进行建模。该提案还涉及（分级）代数的（A-无穷大）变形及其在表示论中的应用。最近，我们与 S. Barmeier 合作开发了一种组合方法来研究具有关系的任何有限箭袋的路径代数的变形理论，这使我们能够轻松计算特定类别的示例。第三个目标旨在理解变形量化下奇点类别的行为，其动机是凯勒猜想。特别是，我们期望循环商奇点的奇点类别与奇点上自然泊松结构的变形量化之间的等价性。第四个目标是根据表面模型描述分级温和代数的 A 无穷大变形直至导出等价。特别是，我们期望一些 A-无穷大变形对应于在表面模型上进行手术。第五个目标是使用我们的组合方法来研究和验证 Stroppel 对扩展 Khovanov 弧代数的猜想，从而产生这些代数的内在形式。