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.

研究建议集中于霍奇柴尔德（Hochschild）以及相关的同事以及更高分类结构（例如A-Infinity类别）的较高代数结构。我的目标是开发适用于代数，表示理论，拓扑和几何形状的各种问题的新技术。在过去的几年中，我一直在研究由单一的Hochschild共同体引起的更高代数结构。我已经证明，单一的霍基柴尔德共同体具有与古典霍奇柴尔德共同体相同的丰富结构：同居中的Gerstenhaber代数和复杂层的B-核代数。凯勒（Keller）提出了这一结果，证明了奇异的霍基柴尔德（Hochschild）同谋与DG奇异性类别的Hochschild共同学是同构的。他还猜想这同构升至B-侵点代数的准同态。我的研究建议中的第一个目标是证明凯勒的猜想上述。为了实现这一目标，与X. -W共同实现。 Chen我们引入了奇异性类别的明确DG增强。在该提案中，我们计划构建B-侵蚀结构对奇异Hochschild共同体学对DG增强的自然作用，该研究将产生B-侵蚀性的质量形态，从而提高Keller的同构。 第二个目标是探索奇异的霍基柴尔德共同体在拓扑和几何学中的进一步应用。 与M. Rivera共同表明，在任何DG Frobenius代数的奇异Hochschild共同体上都有丰富的更高代数结构。在此提案中，我们计划表明，这种较高的结构实际上来自Sullivan DG Prop的自然作用，Sullivan DG Prop最初用于模拟字符串拓扑中的操作。该提案还涉及（分级）代数及其在表示理论的应用（AFINITY）变形。最近，我们与S. barmeier共同开发了一种组合方法来研究与关系的任何有限箭筒的路径代数的变形理论，这使我们能够轻松地计算特定类别的示例类别。第三个目标旨在了解凯勒（Keller）的猜想所激发的变形定量下的奇异性类别的行为。尤其是，我们期望循环商奇异性的奇异性类别与自然泊松结构在奇异性上的变形定量之间存在等效性。第四个目标是描述渐变的温和代数的A-赋值变形，以衍生的等效性，以其表面模型。特别是，我们期望某些A-内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研究和验证Stroppel对扩展的Khovanov Arc代数的猜想，从而产生这些代数的内在形式。