$\text{AssDer}$ 对的扩张、形变与范畴化

带导子的加权Rota-Baxter李代数与结合代数的上同调及形变

Cohomologies and deformations of weighted Rota-Baxter Lie algebras and associative algebras with derivations

该研究引入带导子的加权Rota-Baxter李代数与结合代数配对概念，构建其上同调并探究关联，还应用于形变研究，丰富了相关代数结构理论。

Hecke algebras, finite general linear groups, and Heisenberg categorification

We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type A and finite general linear groups. In this way, we obtain a categorification of the bosonic Fock space. We also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting.

Cohomologies of 3-Lie algebras with derivations

In this paper, we consider a 3-Lie algebra with a derivation (called a 3-LieDer pair). We define cohomology for a 3-LieDer pair with coefficients in a representation. We use this cohomology to study deformations and abelian extensions of 3-LieDer pairs. We give the notion of a 3-Lie2Der pair, which can be viewed as the categorification of a 3-LieDer pair. We show that skeletal 3-Lie2Der pairs are classified by triples given by 3-LieDer pairs, representations and 3-cocycles. We define crossed modules of 3-LieDer pairs and show that there exists a one-to-one correspondence between strict 3-Lie2D…

Hom-associative algebras up to homotopy

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we introduce a strongly homotopy version of hom-associative algebras ($HA_\infty$-algebras in short) on a graded vector space. We describe $2$-term $HA_\infty$-algebras in details. In particular, we study 'skeletal' and 'strict' $2$-term $HA_\infty$-algebras. We also introduce hom-associative $2$-algebras as categorification of hom-associative algebras. The category of $2$-term $HA_\infty$-algebras and the category of hom-associative $2$-algebras are shown to be equivalent. An appr…