Yetter-Drinfeld modules over bosonizations of dually paired Hopf algebras

$C^*$-代数框架下的辫子量子群及其玻色子化

Braided quantum groups and their bosonizations in the $C^*$-algebraic framework

本研究在$C^*$-代数框架下利用乘法酉算子建立辫子量子群一般理论，构造$\bG$上辫子$C^*$-量子群，建立其与带投影$C^*$-量子群的一一对应，推广Radford和Majid的玻色子化构造，并给出复量子平面等实例。

On pointed Hopf algebras over nilpotent groups

We classify finite-dimensional Nichols algebras over finite nilpotent groups of odd order in group-theoretical terms. The main step is to show that the conjugacy classes of such finite groups are either abelian or of type C; this property also holds for finite conjugacy classes of finitely generated nilpotent groups whose torsion has odd order. To extend our approach to the setting of finite GK-dimension, we propose a new Conjecture on racks of type C. We also prove that the bosonization a of Nichols algebra of a Yetter-Drinfeld module over a group whose support is an infinite conjugacy class …

Comodule Algebras and 2-Cocycles over the (Braided) Drinfeld Double

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces…

Representations of $\bar{U}_q s\ell(2|1)$ at even roots of unity

We construct all projective modules of the restricted quantum group $\bar{U}_q s\ell(2|1)$ at an even, $2p$th, root of unity. This $64p^4$-dimensional Hopf algebra is a common double bosonization, $B(X^*)\otimes B(X)\otimes H$, of two rank-2 Nichols algebras $B(X)$ with fermionic generator(s), with $H=Z_{2p}\otimes Z_{2p}$. The category of $\bar{U}_q s\ell(2|1)$-modules is equivalent to the category of Yetter--Drinfeld $B(X)$-modules in $C_ρ={}^H_H\!YD$, where coaction is defined by a universal $R$-matrix $ρ$. As an application of the projective module construction, we find the associative alg…

Examples of finite-dimensional pointed Hopf algebras in positive characteristic

We present new examples of finite-dimensional Nichols algebras over fields of positive characteristic. The corresponding braided vector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups and are analogous to braidings over fields of characteristic zero whose Nichols algebras have finite Gelfand-Kirillov dimension described in arXiv:1606.02521. We obtain nex examples of finite-dimensional pointed Hopf algebras by bosonization with group algebras of suitable finite abelian groups.

Representations of pointed Hopf algebras over S_3

The classification of finite-dimensional pointed Hopf algebras with group S_3 was finished in "The Nichols algebra of a semisimple Yetter-Drinfeld module", arXiv:0803.2430v1 [math.QA], by Andruskiewitsch, Heckenberger and Schneider: there are exactly two of them, the bosonization of a Nichols algebra of dimension 12 and a non-trivial lifting. Here we determine all simple modules over any of these Hopf algebras. We also find the Gabriel quivers, the projective covers of the simple modules, and prove that they are not of finite representation type. To this end, we first investigate the modules o…

Davydov-Yetter 上同调与相对同调代数

Davydov-Yetter cohomology and relative homological algebra

本文通过将 Davydov-Yetter 上同调实现为相对 Ext 群，利用长正合序列、对偶性和 Yoneda 积，为有限张量范畴的无穷小形变提供了新的计算方法，并应用于有限维 Hopf 代数模范畴的具体例子。

Duality for Generalised Differentials on Quantum Groups and Hopf quivers

We study generalised differential structures $Ω^1,d$ on an algebra $A$, where $A\tens A\to Ω^1$ given by $a\tens b\to a d b$ need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs $(Λ^1,ω)$ where $Λ^1$ is a right module and $ω$ a right module map, and the Hopf algebra bicovariant case corresponds to morphisms $ω:A^+\to Λ^1$ in the category of right crossed (or Drinfeld-Radford-Yetter) modules over $A$. When $A=U(g)$ the generalised left-covariant differential structures are classified by cocycles $ω\in Z^1(g,Λ^1)…

On the structure of (co-Frobenius) Hopf algebras

We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra …

关于有限GK维预尼科尔代数和准量子群的分类

On the classification of finite GK-dimensional pre-Nichols algebras and quasi-quantum groups

本文证明了在扭曲Yetter-Drinfeld范畴中非对角对象的预尼科尔代数具有无限GK维，从而完全刻画了该范畴中有限GK维的尼科尔代数，通过bosonization给出了有限GK维的指针余Hopf代数分类。

Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. In order to do that we define the Hochschild cohomology of an algebra in an abelian monoidal category. Then we characterize those algebras which have dimension less than or equal to 1 with respect to Hochschild cohomology. Now let us consider a Hopf algebra A such that its Jacobson radical J is a nilpotent Hopf ideal and H:=A/J is a semisimple algebra. By using our homological results, we prove that the canonical proje…

相对幺半中心与幺半范畴的张量积

The Relative Monoidal Center and Tensor Products of Monoidal Categories

本文建立增广幺半范畴理论，定义平衡双模，构造相对张量积与相对幺半中心并证明其Morita对偶性，还在量子代数等实例中验证了相对中心的范畴等价性。

从对偶张量范畴重构到有限维霍普夫代数的部分对偶拟霍普夫代数

Partially Dualized Quasi-Hopf Algebras Reconstructed from Dual Tensor Categories to Finite-Dimensional Hopf Algebras

研究从对偶张量范畴重构有限维霍普夫代数的部分对偶拟霍普夫代数\((H/B^ + H)^\ast\#B\)，其表示范畴与\(\mathsf{Rep}(H)\)范畴莫里塔等价，统一了经典结果，证明了相关等价性并给出构造拟霍普夫代数的过程。