Corrigendum to Cleft Extensions of Hopf Algebroids

Theorem 2.2 stated a monoidal isomorphism between the comodule categories of two bialgebroids in a Hopf algebroid. The proof of Theorem 2.2 was based on the journal version of Brzeziński (Ann Univ Ferrara Sez VII (NS) 51:15–27, 2005, Theorem 2.6), whose proof turned out to contain an unjustified step. Here we show that all other results in our paper remain valid if we drop unverified Theorem 2.2, and return to an earlier definition of a comodule of a Hopf algebroid that distinguishes between comodules of the two constituent bialgebroids.     

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).     

The weak braided Hopf algebra structure of some Cayley–Dickson algebras

It has been shown by Albuquerque and Majid that a class of unital k-algebras, not necessarily associative, obtained through the Cayley–Dickson process can be viewed as commutative associative algebras in some suitable symmetric monoidal categories. In this note we will prove that they are, moreover, commutative and cocommutative weak braided Hopf algebras within these categories. To this end we first define a Cayley–Dickson process for coalgebras. We then see that the k-vector space of complex numbers, of quaternions, of octonions, of sedenions, etc. fit to our theory, hence they are all monoidal coalgebras as well, and therefore weak braided Hopf algebras.     

Purity and homotopy theory of coalgebras

We construct a combinatorical monoidal model category on simplicial flat cocommutative coalgebras over a Prüfer domain. The cofibrations are the morphisms which are pure as module maps.     

Equivalences of comodule categories for coalgebras over rings

In this article we defined and studied quasi-finite comodules, the cohom functors for coalgebras over rings. Linear functors between categories of comodules are also investigated and it is proved that good enough linear functors are nothing but a cotensor functor. Our main result of this work characterizes equivalences between comodule categories generalizing the Morita-Takeuchi theory to coalgebras over rings. Morita-Takeuchi contexts in our setting is defined and investigated, a correspondence between strict Morita-Takeuchi contexts and equivalences of comodule categories over the involved coalgebras is obtained. Finally, we proved that for coalgebras over QF-rings Takeuchi's representation of the cohom functor is also valid.     

On Gorenstein coalgebras

We investigate the comodule representation category over the Morita-Takeuchi context coalgebra Γ and study the Gorensteinness of Γ. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra Γ and discuss the localization in Gorenstein coalgebras. In particular, we describe its Gabriel quiver and carry out some examples when the Morita-Takeuchi context coalgebra is basic.     

Yetter—Drinfel'd H-Azumaya Monoids in Closed Categories

When C is a symmetric closed category with equalizers and coequalizers and H is a Hopf algebra in C, the category of Yetter—Drinfeld H-modules is a braided monoidal category.We develop a categorical version of the results in (10) constructing a Brauer group BQ(C,H) and studying its functorial properties.     

Generators for comonoids and universal constructions

We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in Agore (Proc Am Math Soc 139:855–863, 2011) on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.     

Duals Invert

Monoidal objects (or pseudomonoids) in monoidal bicategories share many of the properties of the paradigmatic example: monoidal categories. The existence of (say, left) duals in a monoidal category leads to a dualization operation which was abstracted to the context of monoidal objects by Day et al. (Appl Categ Struct 11:229–260, 2003). We define a relative version of this called exact pairing for two arrows in a monoidal bicategory; when one of the arrows is an identity, the other is a dualization. In this context we supplement results of Day et al. (Appl Categ Struct 11:229–260, 2003) (and even correct one of them) and only assume the existence of biduals in the bicategory where necessary. We also abstract recent work of Day and Pastro (New York J Math 14:733–742, 2008) on Frobenius monoidal functors to the monoidal bicategory context. Our work began by focusing on the invertibility of components at dual objects of monoidal natural transformations between Frobenius monoidal functors. As an application of the abstraction, we recover a theorem of Walters and Wood (Theory Appl Categ 3:25–47, 2008) asserting that, for objects A and X in a cartesian bicategory , if A is Frobenius then the category Map(X,A) of left adjoint arrows is a groupoid. Also, the characterization in Walters and Wood (Theory Appl Categ 3:25–47, 2008) of left adjoint arrows between Frobenius objects of a cartesian bicategory is put into our current setting. In the same spirit, we show that when a monoidal object admits a dualization, its lax centre coincides with the centre defined in Street (Theory Appl Categ 13:184–190, 2004). Finally we look at the relationship between lax duals for objects and adjoints for arrows in a monoidal bicategory.     

Generator Coalgebras are not Necessarily Quasi-coFrobenius

We study the problem of whether a coalgebra that generates its category of left (right) comodules is left (right) quasi-coFrobenius or not. We prove it does not hold in general, by giving a method of constructing counterexamples. This gives a negative answer to a question stated in Nastasescu et al. (Algebr Represent Theory 11(2):179–190, 2008). We also prove it is true for monomial pointed coalgebras and we characterize the quivers Q such that \mathbbkQ\mathbb{k}Q admits a monomial subcoalgebra that is left (right) quasi-coFrobenius.     

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.     

余代数上余倾斜余模的结构

研究了余代数上余倾斜余模的结构特征,证明了每个余倾斜余模都可以写成不可分解的两两非同构的余模的直和形式,每个余倾斜余模包含所有的内射不可分解模作为直和项.最后构造了余倾斜余模的两个例子.     

Skew Monoidal Monoids

Skew monoidal categories are monoidal categories with non-invertible “coherence” morphisms. As shown in a previous article, bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod-R in which the unit object is R_R. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present article, we study the one-object case: skew monoidal monoids (SMMs). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids of their own base and are rank 1 free over the base on the source side. We give various equivalent definitions of SMM, study the structure of their (co)module categories, and discuss the possible closed and Hopf structures on a SMM.     

Gorenstein Coalgebras

We study a class of semiperfect coalgebras which generalizes Quasi-co-Frobenius coalgebras (from the point of view of the projective dimension). This class of coalgebras allows us to study a relative concept of injective comodule in terms of the cotensor functor, generalizing a well-known result about coflat comodules.     

The near centre of Doi-Hopf module categoryA M (H) C

It is known that any strict tensor category (C⊗I) determines a braided tensor categoryZ(C), the centre ofC. WhenA is a finite dimension Hopf algebra, Drinfel’d has proved thatZ(_A M) is equivalent to _D(A) M as a braided tensor category, whereA ^M is the left A-module category andD(A) is the Drinfel’d double ofA. For a braided tensor category, the braidC _U,v is a natural isomorphism for any pair of object (U,V) in. If weakening the natural isomorphism of the braidC _U,V to a natural transformation, thenC _U,V is a prebraid and the category with a prebraid is called a prebraided tensor category. Similarly it can be proved that any strict tensor category determines a prebraided tensor category Z∼ (C), the near centre of. An interesting prebraided tensor structure of the Yetter-Drinfel’d category _C*A YD ^C*A given, whereC # A is the smash product bialgebra ofC andA. And it is proved that the near centre of Doi-Hopf module _A M(H) ^C is equivalent to the Yetter-Drinfel’ d _C*A YD ^C*A as prebraided tensor categories. As corollaries, the prebraided tensor structures of the Yetter-Drinfel’d category _A YD ^A , the centres of module category and comodule category are given.     

Tannaka–Kreıˇn reconstruction and a characterization of modular tensor categories

We show that every modular category is equivalent as an additive ribbon category to the category of finite-dimensional comodules of a Weak Hopf Algebra. This Weak Hopf Algebra is finite-dimensional, split cosemisimple, weakly cofactorizable, coribbon and has trivially intersecting base algebras. In order to arrive at this characterization of modular categories, we develop a generalization of Tannaka–Kre?ˇn reconstruction to the long version of the canonical forgetful functor which is lax and oplax monoidal, but not in general strong monoidal, thereby avoiding all the difficulties related to non-integral Frobenius–Perron dimensions. In the more general case of a finitely semisimple additive ribbon category, not necessarily modular, the reconstructed Weak Hopf Algebra is finite-dimensional, split cosemisimple, coribbon and has trivially intersecting base algebras.     

Anchor Maps and Stable Modules in Depth Two

An algebra extension A ∣ B is right depth two if its tensor-square is in the Dress category . We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of two-sided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids and over the centralizer R = C _A (B) are the modules _S R and R _T studied in Kadison (J. Alg. & Appl., 2005, preprint), Kadison (Contemp. Math., 391: 149–156, 2005), and Kadison and Külshammer (Commun. Algebra, 34: 3103–3122, 2006), which provide information about the bialgebroids and the extension (Kadison, Bull. Belg. Math. Soc. Simon Stevin, 12: 275–293, 2005). The anchor maps for the Hopf algebroids in Khalkhali and Rangipour (Lett. Math. Phys., 70: 259–272, 2004) and Kadison (2005, preprint) reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in Van Oystaeyen and Panaite (Appl. Categ. Struct., 2006, in press). We sketch a theory of stable A-modules and their endomorphism rings and generalize the smash product decomposition in Kadison (Proc. Am. Math. Soc., 131: 2993–3002, 2003 Prop. 1.1) to any A-module. We observe that Schneider’s coGalois theory in Schneider (Isr. J. Math., 72: 167–195, 1990) provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. A homomorphism of finite dimensional coalgebras is codepth two if and only if its dual homomorphism of algebras is depth two.      

<Emphasis Type="Italic">β</Emphasis>-character Algebra and a Commuting Pair in Hopf Algebras

Let K be a field. Let H be a finite-dimensional semisimple and cosemisimple K-Hopf algebra. In this paper, we introduce a notion of β-character algebra C _β (H) for each group-like element β in H ^∗. We prove that Radford’s action of the Drinfel’d double D(H) on H _β (see Radford, J. Algebra, 270:670–695, 2003) and the right hit action of the β-character algebra C _β (H) on H _β form a commuting pair. This generalizes an earlier result of Zhu (Proc. Amer. Math. Soc., 125(10):2847–2851, 1997). A K-basis of C _β (H) is given when H is split semisimple. Finally, as an example, we explicitly construct all the simple modules for the Drinfel’d double of the unique 8-dimensional non-commutative and non-cocommutative semisimple Hopf algebra. Presented by S. Montgomery.