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.

     

Invariant hermitian lattices in the steinberg module and their isometry groups

The invariant Hermitian lattices in the Steinberg module of SL₂(q) are described. These lattices are connected with generalized quadratic residue codes over a field of four elements. The isometry groups of in-variant lattices are calculated. In particular, lbdimensional unimodular lattices over Eisenstein numbers with minimum norm 3 and automor- phism group Z₆x PSp₆(3) are obtained.     

广义Hom-李代数的中心不变量

设L是一个广义Hom-李代数, V 是[L,L]的一个H-Hom-李理想. 本文主要研究了L的中心不变量问题. 利用Hopf代数中的方法, 得到了V 的H-不变量包含在L的中心H-不变量中, 这推广了1994年Cohen和Westreich的主要结论.     

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.     

On categories associated to a Quasi-Hopf algebra

A quasi-Hopf algebra H can be seen as a commutative algebra A in the center 𝒵(H-Mod) of H-Mod. We show that the category of A-modules in 𝒵(H-Mod) is equivalent (as a monoidal category) to H-Mod. This can be regarded as a generalization of the structure theorem of Hopf bimodules of a Hopf algebra to the quasi-Hopf setting.     

Construction of Rota-Baxter algebras via Hopf module algebras

We propose the notion of Hopf module algebra and show that the projection onto the subspace of coinvariants is an idempotent Rota-Baxter operator of weight-1. We also provide a construction of Hopf module algebras by using Yetter-Drinfeld module algebras. As an application,we prove that the positive part of a quantum group admits idempotent Rota-Baxter algebra structures.     

Certain categories of modules for twisted affine Lie algebras

In this paper, using generating functions, we study two categories ? and ? of modules for twisted affine Lie algebras g^[σ], which were firstly introduced and studied for untwisted affine Lie algebras by H. -S. Li [Math Z, 2004, 248: 635-664]. We classify integrable irreducible g^[σ]-modules in categories ? and ?, where ? is proved to contain the well-known evaluation modules and ? to unify highest weight modules, evaluation modules and their tensor product modules. We determine also the isomorphism classes of those irreducible modules.     

双单子分配律的R-矩阵

本文讨论了双单子分配律的表示及其R-矩阵结构.设F和G是给定的双单子,刻画了单子双模范畴,并给出了其为辫子范畴的充要条件,由此构造了量子YangBaxter方程的一组新解系.     

Two Results on Brzeziński π-crossed Product

We give the necessary and sufficient conditions for a family of Brzezínski crossed product algebras with suitable comultiplication and counit to be a Hopf π-coalgebra. On the other hand, necessary and sufficient conditions for the Brzeziński π-crossed product A?H to be a coquasitriangular Hopf π-coalgebra are derived, then the category ^A?H ? of the left π-comodules over A?H is braided.     

The tensor product of Gorenstein-projective modules over category algebras

For a finite free and projective EI category, we prove that Gorenstein-projective modules over its category algebra are closed under the tensor product if and only if each morphism in the given category is a monomorphism.     

Adjunctions of Hom and Tensor as Endofunctors of (Bi-)Module Categories Over Quasi-Hopf Algebras

For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor functors ? ? _k V and V ? _k  ? are known to be left adjoint to some kind of Hom-functors as endofunctors of _H 𝕄. The units and counits of adjunctions, in this case, are formally trivial as in the classical case.

In this paper, we generalize this Hom-tensor adjunction for (bi-)module categories over a quasi-Hopf algebra H and show that these (bi-)module categories are biclosed monoidal. However, the units and counits of adjunctions in these generalized cases are not as trivial as in the Hopf algebra case, and they should be modified in terms of the reassociator and the quasi-antipode. Also, if the H-module V is finitely generated and projective as a k-module, we will obtain a generalized form of adjunction between the tensor functors ? ?V and ? ?V* depending on the reassociator and quasi-antipode of H and describe a natural isomorphism between functors ? ?V* and Hom _k (V, ?) explicitly. Furthermore, we consider the special case V = A being an H-module algebra. In this case, each tensor functor will be a monad and its corresponding right adjoint is a comonad. We describe isomorphisms between the (Eilenberg–Moore) module categories over these monads and the (Eilenberg–Moore) comodule categories over their corresponding comonads explicitly.     

Symmetries in yetter-drinfeld categories over weak hopf algebras

This article is devoted to the study of the symmetry in the Yetter-Drinfeld category of a finite-dimensional weak Hopf algebra.It generalizes the corresponding results in Hopf algebras.     

A quasi-Hopf algebra freeness theorem

We prove the quasi-Hopf algebra version of the Nichols-Zoeller theorem: A finite dimensional quasi-Hopf algebra is free over any quasi-Hopf subalgebra.

     

对偶的Hom型拟Hopf代数的余表示和范畴实现

In this paper, we introduce the dual Hom-quasi-Hopf algebra and prove that the comodules category of a(braided) dual Hom-quasi-bialgebra is a monoidal category. Finally,we give a categorical realization of dual Hom-quasi-Hopf algebras.     

Braided doubles and rational Cherednik algebras

We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi-Yetter-Drinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary Yetter-Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double—this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols-Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds in the braided Heisenberg double attached to the corresponding complex reflection group.     

Yetter–Drinfeld Modules over the Hopf–Ore Extension of the Group Algebra of Dihedral Group

Let k be an algebraically closed field of characteristic zero, and D _n be the dihedral group of order 2n, where n is a positive even integer. In this paper, we investigate Yetter-Drinfeld modules over the Hopf-Ore extension A(n, 0) of kD _n . We describe the structures and properties of simple Yetter-Drinfeld modules over A(n, 0), and classify all simple Yetter-Drinfeld modules over A(n, 0).