Simple transitive $2$-representations and Drinfeld center for some finitary $2$-categories

Transitive 2-representations of finitary 2-categories

In this article, we define and study the class of simple transitive $2$-representations of finitary $2$-categories. We prove a weak version of the classical Jordan-Hölder Theorem where the weak composition subquotients are given by simple transitive $2$-representations. For a large class of finitary $2$-categories we prove that simple transitive $2$-representations are exhausted by cell $2$-representations. Finally, we show that this large class contains finitary quotients of $2$-Kac-Moody algebras.

2-Representations and Associated Coalgebra 1-Morphisms for Locally Wide Finitary 2-Categories

We define locally wide finitary 2-categories by relaxing the definition of finitary 2-categories to allow infinitely many objects and isomorphism classes of 1-morphisms and infinite dimensional hom-spaces of 2-morphisms. After defining related concepts including transitive 2-representations in this setting, we provide a new method of constructing coalgebra 1-morphisms associated to transitive 2-representations of locally wide weakly fiat 2-categories, and demonstrate that any such transitive 2-representation is equivalent to a certain subcategory of the category of comodule 1-morphisms over th…

On Simple Transitive 2-representations of Bimodules over the Dual Numbers

We study the problem of classification of simple transitive 2-representations for the (non-finitary) 2-category of bimodules over the dual numbers. We show that simple transitive 2-representations with finitary apex are necessarily of rank 1 or 2, and those of rank 2 are exactly the cell 2-representations. For 2-representations of rank 1, we show that they cannot be constructed using the approach of (co)algebra 1-morphisms. We also propose an alternative definition of (co-)Duflo 1-morphisms for finitary 2-categories and describe them in the case of bimodules over the dual numbers.

有限双范畴的有限双表示

Finitary birepresentations of finitary bicategories

本文将有限2-范畴的有限2-表示理论推广到有限双范畴的有限双表示理论，通过将2-表示的分类结果提升为双表示2-范畴之间的双等价，并证明了双中心化子定理的类似物。

通过局部化研究根方零Nakayama代数上双模的简单传递2-表示

Simple transitive 2-representations of bimodules over radical square zero Nakayama algebras via localization

本文研究了根方零Nakayama代数上双模2-范畴的简单传递2-表示分类问题，通过定义2-表示的局部化并利用构造的泛性质，证明了具有有限顶点的简单传递2-表示均等价于胞腔2-表示的局部化，从而实现了完整分类。

Drinfeld doubles via derived Hall algebras and Bridgeland Hall algebras

Let $\A$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of $\A$ via its derived Hall algebra and Bridgeland Hall algebra of $m$-cyclic complexes.