The bar derived category of a curved dg algebra

表示论中的非齐次Koszul对偶

Non-homogeneous Koszul duality in representation theory

本文研究相伴分次为Koszul的滤过代数，证明其无界导出范畴等价于Positselski弯曲dg对偶代数上内射模同伦范畴的显式商范畴，推广并恢复了Positselski的结果。

通过弯曲 BGG 对应对 Koszul 复形上的厚子范畴进行分类

Classifying thick subcategories over a Koszul complex via the curved BGG correspondence

本文利用弯曲 BGG 对应及完美弯曲 dg 模的厚张量子模分类，刻画了正则环中任意元素列表的 Koszul 复形上有界 dg 模导出范畴的厚子范畴，并恢复 Stevenson、Hopkins–Neeman 等结果。

Cluster categories and duplicated algebras

Let $A$ be a hereditary algebra. We construct a fundamental domain for the cluster category of $A$ inside the category of modules over the duplicated algebra $\bar{A}$ of $A$. We then prove that there exists a bijection between the tilting objects in the cluster category and the tilting $\bar{A}$-modules all of whose non projective-injective indecomposable summands lie in the left part of the module category of $\bar{A}$.

Certain complex representations of $SL_2(\bar{\mathbb{F}}_q)$

We introduce the representation category $\mathscr{C}({\bf G})$ for a connected reductive algebraic group ${\bf G}$ which is defined over a finite field $\mathbb{F}_q$ of $q$ elements. We show that this category has many good properties for ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$. In particular, it is an abelian category and a highest weight category. Moreover, we classify the simple objects in $\mathscr{C}({\bf G})$ for ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$.

Odd annular Bar-Natan category and gl(1|1)

We introduce two monoidal supercategories: the odd dotted Temperley-Lieb category $\mathcal{T\!L}_{o,\bullet}(δ)$, which is a generalization of the odd Temperley-Lieb category studied by Brundan and Ellis, and the odd annular Bar-Natan category $\mathcal{BN}_{\!o}(\mathbb{A})$, which generalizes the odd Bar-Natan category studied by Putyra. We then show there is an equivalence of categories between them if $δ=0$. We use this equivalence to better understand the action of the Lie superalgebra $\mathfrak{gl}(1|1)$ on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and…

Cartan型李代数$\bar{S}_2$上的Whittaker模范畴

The category of Whittaker modules over the Cartan Type Lie algebra $\bar{S}_2$

研究$\mathbb{C}^2$上常散度多项式向量场李代数$\bar{S}_2$的Whittaker模，通过等价于抛物子代数有限维模范畴分类了所有单Whittaker模，并建立了与结合代数$H_{\mathbf{1}}$有限维模范畴的等价。