Fiat 2-范畴的2-表示理论

本项目主要研究fiat 2-范畴的2-表示理论。它被认为是2-版本的带对合的有限维代数。平行于经典表示论，在2-表示理论中存在弱的Jordan-Hölder定理并且“单表示”为单可迁2-表示。2-表示理论中的一个基本问题就是单可迁2-表示的分类。通过应用胞腔结构，这一基本问题可以被分解为一些子的分类问题。其上胞腔结构推广了半群中的Green关系和Coxeter群上Hecke代数的Kazhdan-Lusztig胞腔。另一方面，有限维代数的中心在经典表示论中起着至关重要的作用。Meir和Szymik引入了抽象2-范畴的中心，称为Drinfeld中心，推广了幺半范畴上的相应概念。本项目致力于分类fiat 2-范畴的单可迁2-表示和刻画它的Drinfeld中心。作为应用，申请人将进一步研究有限Coxeter群上余不变代数的Soergel双模生成的2-范畴。

The proposal mainly studies 2-representation theory of fiat 2-categories, viewed as 2-analogues of finite dimensional algebras with involutions. Parallel to classical representation theory, we have the weak Jordan-Hölder theorem in the 2-representation theory and the "simples" are played by the so-called simple transitive 2-representations. A fundamental problem in 2-representation theory is the classification problem of simple transitive 2-representations, which can be subdivided into several smaller classification problems by taking advantage of the so-called cell structures. The latter generalize Green’s relations for semigroups and the well-known Kazhdan-Lusztig cells on the Hecke algebras of Coxeter groups. On the other hand, the centers of finite dimensional algebras play a central role in the classical representation theory. Meir and Szymik introduced the centers of abstract 2-categories, called the Drinfeld centers as a generalization of the ones for monoidal categories. The proposal aims to classify simple transitive 2-representations for fiat 2-categories and characterize the Drinfeld centers. As an application, we would like to further study the 2-category of Soergel bimodules over the coinvariant algebra of finite Coxeter groups.