Yangians and Cohomological Hall algebras of curves

在2022财年期间，我专注于在一般框架下定义双(2维)上同调霍尔代数(COHA)和范畴霍尔代数(CatHAs)。给定DG范畴C(满足某些技术条件)的固定t-结构的中心上的一个扭转对(T，F)，在[arxiv：2207.08926]中，Diaconescu，Porta，和我引入了一个一般的形式主义来联系一个COHA(分别.Catha)与左、右模块(分别为范畴化模)与F.全杨安(分别范畴杨安)被定义为T的么半群(分别为结合)由这两个模结构生成的代数。这一框架给出了Negut在[Arxiv：1703.02027]中引入的光滑射影复曲面S的Ding-Iohara-Miki代数的一个范畴以及它的一个推广，包括“沿曲线的Hecke修改”算子；它允许定义S上PT稳定对的模空间的同调、K-理论等作用。在[arxiv：2303.17154]中，Diaconescu，Porta和我利用Motivic Hall代数研究了空间曲线奇点上PT稳定对的模空间的几何和点的Hilbert格式，给出了当空间曲线奇点局部完全相交时，它们的欧拉数的母函数的显式公式。最后，Diaconescu，Porta，Schiffmann，Vasserot，并且我正在完成关于一类ADE奇点的最小分辨的COHA的描述的项目，关于同一类型的仿射杨安。

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.