Higher structures and deformations in representation theory

该研究计划的重点是从Hochschild产生的高级代数结构，以及相关的上同调和高级范畴结构，如A-无穷范畴。我的目标是开发适用于代数、表示论、拓扑学和几何学中广泛问题的新的先进技术。在过去的几年里，我一直在研究奇异Hochschild上同调所产生的高等代数结构。我已经证明了奇异Hochschild上同调被赋予了与经典Hochschild上同调相同的丰富结构：一个Gerstenhaber代数的上同调和一个B-无穷代数的复水平。根据这个结果，凯勒证明了奇异Hochschild上同调同构于dg奇异范畴的Hochschild上同调。他还指出，这种同构升降机准同构的B-无限代数。我的研究计划的第一个目标是证明上述凯勒猜想。为了实现这一点，与X。-W.我们引入了奇点范畴的一个显式dg增强。在这个提议中，我们计划在这个dg增强上构造一个B-无穷结构对奇异Hochschild上同调的自然作用，这将产生一个提升Keller同构的B-无穷拟同构。 第二个目标是探索奇异Hochschild上同调在拓扑学和几何学中的进一步应用。 与M。里维拉证明了任意dg Frobenius代数的奇异Hochschild上同调都有丰富的高等代数结构。在这个提议中，我们计划表明，这种更高的结构实际上来自沙利文dg PROP的自然动作，最初用于模拟字符串拓扑中的操作。这个建议还涉及（分次）代数的（A-无穷）变形及其在表示论中的应用。最近，与S。Barmeier，我们已经开发了一种组合方法来研究变形理论的道路代数的任何有限的关系，这使我们能够很容易地计算特定类别的例子。第三个目标旨在了解变形量子化下的奇异性类别的行为，由凯勒猜想的动机。特别是，我们期望一个循环商奇异性和奇异性的自然泊松结构的变形量化的奇异性类别之间的等价性。第四个目标是描述A-无穷变形的分次温柔代数导出等价，在他们的表面模型。特别地，我们期望一些A-无穷变形对应于在表面模型上做手术。第五个目标是用我们的组合方法研究和验证扩展Khovanov弧代数的Stroppel猜想，从而得到这些代数的内在形式。

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.