Yangians and Cohomological Hall algebras of curves

在2022财年，我专注于在一般框架中定义双（二维）上同调霍尔代数（COHAs）和分类霍尔代数（CatHAs）。给定一类C（满足一定技术条件）的固定T -结构中心上的扭转副（T，F），在[arXiv:2207.08926]中，Diaconescu, Porta， I引入了关联COHA的一般形式。CatHA)到T，连同一个左右模块(参见。分类模组)与F.完整的Yangian(参见；定义T的类Yangian为单轴(p。由这两个模块结构生成的结合代数。例如，该框架提供了Negut在[arXiv:1703.02027]中引入的光滑射影复曲面S的Ding-Iohara-Miki代数的分类及其包含“沿曲线Hecke修正”算子的推广；在[arXiv:2303.17154]中，Diaconescu, Porta， and I研究了空间曲线奇点上PT稳定对的模空间和Hilbert格式的模空间的几何，利用动力Hall代数，给出了空间曲线奇点为局部完全相交时它们的欧拉数的生成函数的显式公式。最后，Diaconescu、Porta、Schiffmann、Vasserot和我正在完成一个项目，这个项目是关于用相同类型的仿射Yangian描述ADE奇点最小分辨率的COHA。

During the FY2022, I focused on the definition of double (2-dimensional) cohomological Hall algebras (COHAs) and categorified ones (CatHAs) in a general framework. Given a torsion pair (T,F) on the heart of a fixed t-structure of a dg category C (satisfying certain technical conditions), in [arXiv:2207.08926], Diaconescu, Porta, and I introduce a general formalism to associate a COHA (resp. CatHA) to T together with a left and right module (resp. categorified module) associated to F. The full Yangian (resp. categorified Yangian) of T is defined as the monoidal (resp. associative) algebra generated by these two module structures. This framework provides for example a categorification of the Ding-Iohara-Miki algebra of a smooth projective complex surface S introduced by Negut in [arXiv:1703.02027] and a generalization of it including operators of "Hecke modifications along curves"; it allows the definition of actions on the homology, K-theory, etc, of the moduli space of PT stable pairs on S.In [arXiv:2303.17154], Diaconescu, Porta, and I studied the geometry of the moduli space of PT stable pairs and Hilbert schemes of points on a space curve singularity, using motivic Hall algebras, providing an explicit formula of the generating function of their Euler numbers when the space curve singularity is locally complete intersection.Finally, Diaconescu, Porta, Schiffmann, Vasserot, and I are finalizing the project about a description of the COHA of the minimal resolution of a type ADE singularity in terms of the affine Yangian of the same type.