Abel匹配对的上同调与Kac序列

关于可平行流形上向量场的上同调

On the cohomology of vector fields on parallelizable manifolds

本文对每个可平行化的光滑紧致流形$M$确定了$M$上光滑向量场的李代数$V_M$的上同调空间$H^2(V_M,\barΩ^p_M)$，其值在模$\barΩ^p_M = Ω^p_M/dΩ^{p-1}_M$中. $p=1$的情况特别有趣，因为规范代数$C^\infty (M,k)$具有以中心$\barΩ^1_M$为中心的泛中心扩张，推广了仿射Kac-Moody代数。二阶上同调$H^2(V_M, \barΩ^1_M)$将$V_M$的半直积与通用中心扩张$C^\infty (M,k) \oplus \barΩ^1_M$的扭曲进行分类。

Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a $1$-$1$ correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the e…

The bicrossed products of $H_4$ and $H_8$

Let $H_4$ and $H_8$ be the Sweedler's and Kac-Paljutkin Hopf algebras, respectively. In this paper we prove that any Hopf algebra which factorizes through $H_8$ and $H_4$ (equivalently, any bicrossed product between the Hopf algebras $H_8$ and $H_4$) must be isomorphic to one of the following four Hopf algebras: $H_8 \otimes H_4, H_{32,1}, H_{32,2}, H_{32,3}$. The set of all matched pair $(H_8, H_4, \triangleright, \triangleleft)$ is explicitly described, and then the associated bicrossed products is given by generators and relations.

Centers of KLR algebras and cohomology rings of quiver varieties

Attached to a weight space in an integrable highest weight representation of a simply-laced Kac-Moody algebra $\mathfrak{g}$, there are two natural commutative algebras: the cohomology ring of a quiver variety and the center of a cyclotomic KLR algebra. In this note, we describe a natural geometric map between these algebras in terms of quantum coherent sheaves on quiver varieties. The cohomology ring of an algebraic symplectic variety can be interpreted as the Hochschild cohomology of a quantization of this variety in the sense of Bezrukavnikov and Kaledin. On the other hand, cyclotomic KLR a…

On Hom-pre-Lie bialgebras

In this paper we introduce the notion of Hom-pre-Lie bialgebra in the general framework of the cohomology theory for Hom-Lie algebras. We show that Hom-pre-Lie bialgebras, standard Manin triples for Hom-pre-Lie algebras and certain matched pairs of Hom-pre-Lie algebras are equivalent. Due to the usage of the cohomology theory, it makes us successfully study the coboundary Hom-pre-Lie bialgebras. The notion of Hom-s-matrix is introduced, by which we can construct Hom-pre-Lie bialgebras naturally. Finally we introduce the notions of Hom-O-operators on Hom-pre-Lie algebras and Hom-L-dendriform al…

Hopf-Galois 扩张的 Hochschild 上同调上的杯积

Cup Products on Hochschild Cohomology of Hopf-Galois Extensions.pdf

本文构造显式链映射，在温和假设下证明Hopf-Galois扩张的Hochschild上同调与H-不变子代数同构，推广群作用经典结果，并计算量子(-1)-平面与Kac-Paljutkin Hopf代数smash积的上同调及杯积。

Invariants of Lie algebras extended over commutative algebras without unit

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.

Nijenhuis李代数与（一般）Nijenhuis李双代数的上同调理论

Cohomology theory of Nijenhuis Lie algebras and (generic) Nijenhuis Lie bialgebras

该文构建Nijenhuis李代数的上同调理论以刻画同时形变，拓展研究至一般Nijenhuis李双代数，引入匹配对与Manin三元组证明其等价性，并关联容许经典Yang-Baxter方程的反对称解。

双泊松（顶点）代数上同调

Double Poisson (vertex) algebra cohomology

本文去除了双泊松括号由非交换双向量定义的前提，构建了完备的双泊松代数上同调理论，并推广到拟泊松和规范泊松情形；同时发展了双泊松顶点代数上同调理论，证明了其在表示函子下与交换版本兼容。

n-李代数的扩张结构

Extending Structures for $n$-Lie Algebras

本文从三个角度重新研究n-李代数的上同调理论，重点探讨其扩张结构，得到统一乘积判据及相关特例，还研究了分解、形变映射等问题。

同伦共形代数

Homotopy conformal algebras

本文引入 $A_\infty$-共形代数与结合共形 $2$-代数，证明同伦迁移定理及二者范畴等价，并探讨其与 $L_\infty$-共形代数的关系。

Nijenhuis算子与mock-Lie双代数

Nijenhuis operators and mock-Lie bialgebras

本文通过推广匹配对、Manin三元组等概念，刻画了Nijenhuis mock-Lie双代数，并研究了其形式形变、阿贝尔扩张及与mock-Lie-Yang-Baxter方程的联系。