Projections of weak braided Hopf algebras

In this paper we study the projections of weak braided Hopf algebras using the notion of Yetter-Drinfeld module associated with a weak braided Hopf algebra. As a consequence, we complete the study ofthe structure of weak Hopf algebras with a projection in a braiding setting obtaining a categorical equivalencebetween the category of weak Hopf algebra projections associated with a weak Hopf algebra H living in abraided monoidal category and the category of Hopf algebras in the non-strict braided monoidal cate...     

群Yetter-Drinfel''d范畴中的Hopf代数

In this note we first show that if H is a finite-dimensional Hopf algebra in a group Yetter-Drinfel＇d category L^LyD（π） over a crossed Hopf group-coalgebra L, then its dual H^＊ is also a Hopf algebra in the category L^LyD（π）. Then we establish the fundamental theorem of Hopf modules for H in the category L^LyD（π）.     

正则乘子Hopf代数上Yetter-Drinfel''d模范畴中的自同构代数

本文研究了正则乘子Hopf代数上Yetter-Drinfel''d模范畴中自同构代数的问题.利用乘子Hopf代数以及同调代数理论中的方法,获得了Yetter-Drinfel''d模范畴中两个自同构代数是同构的结果,推广了Panaite等人在Hopf代数中的结果.     

On the category of hyper MV‐algebras

In this paper we study the category of hyper MV‐algebras and we prove that it has a terminal object and a coequalizer. We show that Jia's construction can be modified to provide a free hyper MV‐algebra by a set. We use this to show that in the category of hyper MV‐algebras the monomorphisms are exactly the one‐to‐one homomorphisms. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalizations of the Sweedler Dual

As left adjoint to the dual algebra functor, Sweedler’s finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler’s construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.     

Geometric partial comodules over flat coalgebras in Abelian categories are globalizable

The aim of this paper is to prove the statement in the title. As a by-product, we obtain new globalization results in cases never considered before, such as partial corepresentations of Hopf algebras. Moreover, we show that for partial representations of groups and Hopf algebras, our globalization coincides with those described earlier in literature. Finally, we introduce Hopf partial comodules over a bialgebra as geometric partial comodules in the monoidal category of (global) modules. By applying our globalization theorem we obtain an analogue of the fundamental theorem for Hopf modules in this partial setting.     

Finite Presentability and the Homological Type FP m for a Class of Hopf Algebras

We define and study the property finite presentability in the category  of Hopf algebras that are smash product of universal enveloping algebra of a Lie algebra by a group algebra. We show that for such Hopf algebras finite presentability is equivalent with finite presentability as an associative k-algebra.     

Nichols algebras over weak Hopf algebras

In this paper, we study a Yetter-Drinfeld module V over a weak Hopf algebra H.Although the category of all left H-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Drinfeld modules V, we construct Nichols algebra B(V) over the weak Hopf algebra H, and a series of weak Hopf algebras. Some results of [8] are generalized.     

三角Hopf余模范畴上的张量余代数

在三角Ｈｏｐｆ代数余模范畴上研究张量余代数．主要给出三角Ｈｏｐｆ代数余模范畴上的张量余代数的结构．     

辫子Hopf代数的积分

本文引进了无限维辫子Hopf代数日的忠实拟对偶H~d和严格拟对偶H~(d′)．证明了每个严格拟对偶H~(d′)是一个H-Hopf模．发现了H~d的极大有理H~d-子模H~(drat)与积分的关系,即：H~(drat)≌∫_(H~d)~l■H．给出了在Yetter-Drinfeld范畴(_B~ByD,C)中的辫子Hopf代数的积分的存在性和唯—性．     

Hopf Algebra Extensions of Group Algebras and Tambara-Yamagami Categories

We determine the structure of Hopf algebras that admit an extension of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension < 36 is necessarily group-theoretical; thus 36 is the smallest possible dimension where a non group-theoretical example occurs.     

Cleft extensions in braided categories

Majid in [14] and Bespalov in [2] obtain a braided interpretation of Radford’s theorem about Hopf algebras with projection ([19]). In this paper we introduce the notion of H-cleft comodule (module) algebras (coalgebras) for a Hopf algebra H in a braided monoidal category, and we characterize it as crossed products (coproducts). This allows us give very short proofs for know results in our context, and to introduce others stated for the category of R-modules about of Hopf algebra extensions. In particular we give a proof of the result by Bespalov [2] for a braided monoidal category with co(equalizers).     

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.     

The formal theory of Hopf algebras Part II: The case of Hopf algebras

Abstract

The category Hopf_R of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category Hopf_R is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then Hopf_R is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of Hopf_R. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist.     

An Algebraic Framework for Group Duality

A Hopf algebra is a pair (A, Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, Δ) is canonically isomorphic with the original multiplier Hopf algebra (A, Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.     

关于$\omega$-Smash余积Hopf代数的$(f,\omega)$相容对$(B,,H)$

In this paper we introduce the notion of （f,ω）-compatible pair （B,H）, by which we construct a Hopf algebra in the category HHYD of Yetter-Drinfeld H-modules by twisting the comultiplication of B. We also study the property of ω-smash coproduct Hopf algebras Bω H.     

Hopf monads on monoidal categories

We define Hopf monads on an arbitrary monoidal category, extending the definition given in Bruguières and Virelizier (2007) [5] for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode.Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid.Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C. The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford–Majid bosonization of Hopf algebras).We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedler?s Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).     

Galois-type correspondences for Hopf Galois extensions

In order to obtain a Galois correspondence between intermediate subalgebras of a Hopf-Galois extension and corresponding Hopf subalgebras, we have to reconstruct such Hopf algebras from the given Hopf-Galois extension. The construction of a suitable Hopf algebra associated to a commutativeH-Galois extension is provided; the results are applicable to field extensions.This author is supported by a grant of the European Liaisons Committee.     

DG-Modules vs. Relative Hopf Modules

In this note we show the interlacing between homological algebra and the category of relative Hopf modules. We show that many well-known concepts related with differential graded algebras can be performed as purely Hopf algebraic phenomenons.     

Pointed Hopf Algebras—from Enveloping Algebras to Quantum Groups and Beyond

A fundamental example of a pointed Hopf algebra is the universal enveloping algebra of a Lie algebra. This example finds its generalization in quantum groups. The class of pointed Hopf algebras is rather extensive and has been the subject of intense study. We recount some of the basic ideas in the development of the theory. A fundamental structure in the general theory is an analog of the enveloping algebra in a certain category, the Nichols algebra.