Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras,Weak Hopf Algebras,and Braided Hopf Algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra, or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v: H → B. Then we can define an object B^co(H), which is a left-left Yetter–Drinfeld module over H, having extra properties that allow to make a smash product B^co(H)#H, which is an H-bicomodule algebra, isomorphic to B.     

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

For a quasi-Hopf algebra H, a left H-comodule algebra  and a right H-module coalgebra C we will characterize the category of Doi–Hopf modules ^C ?(H) in terms of modules. We will also show that for an H-bicomodule algebra  and an H-bimodule coalgebra C the category of generalized Yetter–Drinfeld modules (H) ^C is isomorphic to a certain category of Doi–Hopf modules. Using this isomorphism we will transport the properties from the category of Doi–Hopf modules to the category of generalized Yetter–Drinfeld modules.     

量子环面上一类结合代数的表示

本文研究了与量子环面CQ[x±1,y±1](见[6])有关的一类结合代数的表示.该结合代数的表示与广义仿射李代数(见[1])的表示理论密切相关.本文推广了[5]的部分结果.     

The Coradical Filtration for Drinfeld Double Ringel-Hall Algebras

Abstract

Let 𝒟(Λ) be the Drinfeld double Ringel-Hall algebra with Λ being any finite dimensional hereditary algebra over a finite field k. We determine the coradical filtration for 𝒟(Λ). As an application, we describe the group of Hopf algebra automorphisms of the Drinfeld double Ringel composition algebra of Λ.     

Graded Quantum Groups and Quasitriangular Hopf Group-Coalgebras

ABSTRACT

Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GL _n (𝕂). In particular, we define the graded quantum groups, which are Hopf π-coalgebras for π = ?[[h]] ^l and generalize the Drinfeld-Jimbo quantum enveloping algebras.     

A Categorical Approach to Turaev's Hopf Group-Coalgebras

We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier's version of the Fundamental Theorem for Hopf algebras. We introduce Yetter–Drinfeld modules over Hopf group-coalgebras using the center construction.     

REPRESENTATIONS OF A LOOP LIE ALGEBRA ASSOCIATED WITH QUANTUM PLANE

In this article,some modules over a loop Lie algebra associated to quantum plane are constructed.The isomorphism classes among these modules are also determined.     

HOPF代数的右伴随作用构造的上循环模(英文)

本文研究了上循环模,对于特征为O的域k上满足S~2=id_H的Hopf代数H,和左H-模代数A,利用日的右伴随作用以及H在A上的模作用,构造了上循环模(C)_H~#(A),并且证明了由H的右伴随作用和左伴随作用分别诱导的上循环模(C)_H~(#)(A)和(C)_H~(#)(A)足同构的.     

Two-sided Two-cosided Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

For a quasi-Hopf algebra H, an H-bicomodule algebra and an H-bimodule coalgebra C we will show that the category of two-sided two-cosided Hopf modules is equivalent to the category of right–left generalized Yetter–Drinfeld modules . Using alternative versions of this result we will recover the category isomorphism between the categories of left–left and left–right Yetter–Drinfeld modules over a quasi-Hopf algebra.      

REPRESENTATIONS OF THE TWO PARAMETER DEFORMATION OF THE VIRASORO ALGEBRA

This paper constructs a class of Harish-Chandra modules with multiplicity ≤1 of the two parameter deformation of Virasoro algebra and proves a classification theorem.     

Higher Indicators for the Doubles of Some Totally Orthogonal Groups

We investigate the indicators for certain groups of the form ? _k  ? D _l and their doubles, where D _l is the dihedral group of order 2l. We subsequently obtain an infinite family of totally orthogonal, completely real groups which are generated by involutions, and whose doubles admit modules with second indicator of ?1. This provides us with answers to several questions concerning the doubles of totally orthogonal finite groups.     

Self-dual weak Hopf algebras

In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually simple forms in the sense of E. L. Green and E. N. Morcos. We also give the definition of self-dual weak Hopf quiver and apply these types of quivers to classify the finite- dimensional self-dual semilattice graded weak Hopf algebras. Finally, we prove partially the conjecture given by N. Andruskiewitsch and H.-J. Schneider in the case of finite-dimensional pointed semilattice graded weak Hopf algebra H when grH is self-dual.     

Hopf modules and the double of a quasi-Hopf algebra

We give a different proof for a structure theorem of Hausser and Nill on Hopf modules over quasi-Hopf algebras. We extend the structure theorem to a classification of two-sided two-cosided Hopf modules by Yetter-Drinfeld modules, which can be defined in two rather different manners for the quasi-Hopf case. The category equivalence between Hopf modules and Yetter-Drinfeld modules leads to a new construction of the Drinfeld double of a quasi-Hopf algebra, as proposed by Majid and constructed by Hausser and Nill.

     

Green rings of Drinfeld doubles of Taft algebras

Abstract

In this article, we investigate the representation rings (or Green rings) of the Drinfeld doubles of the Taft algebras. It is shown that these Green rings are commutative rings generated by infinitely many elements subject to certain relations. The generators together with the subjecting relations are given. The stable Green rings of of the Drinfeld doubles of the Taft algebras are also described.     

一类Hopf路余代数

设F是域,G是3阶循环群,Q是群G的箭图.借助于模范畴的等价性,给出了Hopf路余代数FQ~c的所有结构分类,并给出了FQ~c的子Hopf代数FG[FQ_1~c]的结构.     

The Structure of Weak Hopf Algebras Corresponding to U q (sl 2)

The structure of weak Hopf algebras corresponding to U _q (sl ₂) are classified by their algebra structure and coalgebra structure. The algebra structure of weak Hopf algebras corresponding to U _q (sl ₂) can be written as the direct sum of U _q (sl ₂) and an algebra of polynomials. The coalgebra structure of weak Hopf algebras corresponding to U _q (sl ₂) are classified by their Ext quiver. There are four types of such structures.     

Finite Majid Algebras Over the Klein Group

We provide a classification of finite-dimensional connected coradically graded pointed Majid algebras, generated in degrees 0 and 1, over the Klein group on the field of complex numbers.