The naive approach for constructing the derived category of a $d$-abelian category fails

在 Borel 量子环代数的范畴 $\mathcal{O}$ 中构造 $R$-矩阵的函子

A functor for constructing $R$-matrices in the category $\mathcal{O}$ of Borel quantum loop algebras

本文通过定义正合函子 $\mathcal{F}_q$ 并利用已有的辫子结构，解决了为任意非扭曲量子环代数 Borel 子代数的范畴 $\mathcal{O}$ 构造 $R$-矩阵的问题，并推导出了 Grothendieck 环的新关系及同构的函子解释。

Internally Calabi-Yau algebras and cluster-tilting objects

We describe what it means for an algebra to be internally d-Calabi-Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1)-cluster-tilting objects in certain stably (d-1)-Calabi-Yau Frobenius categories, as observed by Keller-Reiten. We show that an internally d-Calabi-Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d-1)-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger 'bimodule' internally d-Calabi-Yau condition, this Frobenius category is stably …

第二德尔安尼类别

The second Delannoy category

本文研究了第二德尔安尼范畴的结构，并构造了其交换版本，同时确定了该范畴的两个局部交换包络。

由簇倾斜代数构造倾斜代数

Constructing tilted algebras from cluster-tilted algebras

任何簇倾斜代数都是倾斜代数的关系扩张。给出了一种方法，即给定一个簇倾斜对象在簇范畴的Auslander-Reiten环中的分布，构造所有关系扩张为该簇倾斜对象的自同态环的倾斜代数.

Representing the big tilting sheaves as holomorphic Morse branes

We introduce Morse branes in the Fukaya category of a holomorphic symplectic manifold, with the goal of constructing tilting objects in the category. We give a construction of a class of Morse branes in the cotangent bundles, and apply it to give the holomorphic branes that represent the big tilting sheaves on flag varieties.

通过对称幺半范畴的插值构造新的开闭拓扑量子场论（TQFT）

Constructing new open-closed TQFTs from the interpolation of symmetric monoidal categories

本研究基于Lauda和Pfeiffer关于二维开闭TQFT与KFAs等价的结论，通过任意标量序列构建含KFA对象的对称幺半范畴，确定了生成半单阿贝尔范畴的序列条件，得到插值特定KFA自同构群表示范畴的新范畴族。

Recollements arising from cotorsion pairs on extriangulated categories

This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category. As an application, this result generalizes the work by Chen-Liu-Yang in a triangulated case. Moreover, it highlights new phenomena when it applied to an exact category. Finally, we give some applications to illustrate our main results. In particular, we obtain the Krause's recollement whose the proofs are both elementary and very general.

The Hochschild Complex of a Finite Tensor Category

Modular functors, i.e. consistent systems of projective representations of mapping class groups of surfaces, have been constructed for non-semisimple modular categories already decades ago. Concepts from homological algebra have not been used in this construction although it is an obvious question how they should enter in the non-semisimple case. In the present paper, we elucidate the interplay between the structures from topological field theory and from homological algebra by constructing a homotopy coherent projective action of the mapping class group $\text{SL}(2,\mathbb{Z})$ of the torus …

Prime spectra of abelian 2-categories and categorifications of Richardson varieties

We describe a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category. This provides a noncommutative version of Balmer's prime spectrum of a tensor triangulated category. These notions are based on containment conditions in terms of thick subcategories of an abelian category and thick ideals of an abelian 2-category. We prove categorical analogs of the main properties of noncommutative prime spectra. Similar notions, starting with Serre subcategories of an abelian category and Serre ideals of an abelian 2-category, are developed. They are linked…

Q-形模范畴上的模型结构

Model structures on the category of Q-shaped modules

该研究开发了从AMod的余挠对构造Q-形模范畴上阿贝尔模型结构的方法，在Q无环时可得到投射模型结构，并将其应用于微分左A-模范畴，获得了相容的诱导余挠对及阿贝尔模型结构。

通用几何簇代数

Universal geometric cluster algebras

本文研究了基于交换矩阵B的几何簇代数及其系数特化，探讨了通用几何系数的构造，通过突变风扇F_B与g向量的关系，讨论了有限类型和仿射类型的构造方法。

阿贝尔范畴图表上的表示 I：全局结构与同调对象

Representations over diagrams of abelian categories I: Global structure and homological objects

本文系统研究阿贝尔范畴图表上的表示，探讨表示范畴的Grothendieck结构、重要函子与伴随关系，并刻画特殊同调对象，为后续工作奠定基础。

可定义函子与布朗-阿德斯可代表性

Definable functors and Brown--Adams representability

本文研究了环的导出范畴中布朗-阿德斯可代表性的条件，通过可定义函子的同构转换，证明了纯全局维数的关系，并给出了茶杯猜想的新证明及协调环的生成假设新结论。