Breuil–Kisin Modules and Hopf Orders in Cyclic Group Rings

For K, a finite extension of ? _p with ring of integers R, we show how Breuil–Kisin modules can be used to determine Hopf orders in K-Hopf algebras of p-power dimension. We find all cyclic Breuil–Kisin modules and use them to compute all of the Hopf orders in the group ring KΓ where Γ is cyclic of order p or p ². We also give a Laurent series interpretation of the Breuil–Kisin modules that give these Hopf orders.     

Fundamental Theorems of Weak Doi–Hopf Modules and Semisimple Weak Smash Product Hopf Algebras

Abstract

This paper, mainly gives a Fundamental Theorem of weak Doi–Hopf modules, which is not only generalizes the Fundamental Theorem of weak Hopf modules but also generalizes the Fundamental Theorem of relative Hopf modules. Moreover, it gives a sufficient and necessary condition for weak smash product algebras to be weak bialgebras, and a sufficient condition for weak smash product algebras to be semisimple weak Hopf algebras.     

THE FUNCTOR AND THE PROJECTIVITY AND INJECTIVITY OF RELATIVE HOPF MODULES

ABSTRACT

Let H be a Hopf algebra over a field k, and A a commutative right H-comodule algebra. This paper is concerned with homological algebra for Hopf modules. We investigate the projectivity, injectivity, and minimal injective resolutions of relative Hopf modules by using the functor HOM.     

Cibils–Rosso's Theorem for Quantum Groupoids

Abstract

We generalize the Cibils–Rosso's theorem for categories of Sweedler's Hopf bimodules to the one for categories of weak entwined bimodules. We show that the weak entwined bimodules are modules over a certain algebra. Our best results are attained for categories of weak Hopf bimodules over quantum groupoids (weak Hopf algebras), as special cases of weak Doi–Hopf bimodules.     

Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras

Abstract

In this paper we construct a cylindrical module A ? ? for an ?-comodule algebra A, where the antipode of the Hopf algebra ? is bijective. We show that the cyclic module associated to the diagonal of A ? ? is isomorphic with the cyclic module of the crossed product algebra A ? ?. This enables us to derive a spectral sequence for the cyclic homology of the crossed product algebra. We also construct a cocylindrical module for Hopf module coalgebras and establish a similar spectral sequence to compute the cyclic cohomology of crossed product coalgebras.     

Algebras Graded by Discrete Doi–Hopf Data and the Drinfeld Double of a Hopf Group-Coalgebra

We study Doi–Hopf data and Doi–Hopf modules for Hopf group-coalgebras. We introduce modules graded by a discrete Doi–Hopf datum; to a Doi–Hopf datum over a Hopf group coalgebra, we associate an algebra graded by the underlying discrete Doi–Hopf datum, using a smash product type construction. The category of Doi–Hopf modules is then isomorphic to the category of graded modules over this algebra. This is applied to the category of Yetter–Drinfeld modules over a Hopf group coalgebra, leading to the construction of the Drinfeld double. It is shown that this Drinfeld double is a quasitriangular ${\mathbb{G}}$ -graded Hopf algebra.     

Bialgebra Cyclic Homology with Coefficients

We show that one can extend the definition of Hopf cyclic homology with coefficients such that one can use bialgebras and a larger class of coefficient module/co-modules. With the help of this extension, we calculate the bialgebra cyclic homology of U_q() the quantum deformation of an arbitrary semi-simple Lie algebra and (N) the Hopf algebra of foliations of codimension N, with several coefficient modules.     

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.     

Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras

We extend the Larson–Sweedler theorem [Amer. J. Math. 91 (1969) 75] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements, we derive the Radford formula [Amer. J. Math. 98 (1976) 333] for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra A^T of the underlying weak Hopf algebra A.     

Multiplier Hopf Monoids

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele’s definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved — in an appropriate, multiplier-valued sense — which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved — which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.     

Formal Hopf algebra theory I: Hopf modules for pseudomonoids

In this article we develop some of the basic constructions of the theory of Hopf algebras in the context of autonomous pseudomonoids in monoidal bicategories. We concentrate on the notion of Hopf modules. We study the existence and the internalisation of this notion, called the Hopf module construction. Our main result is the equivalence between the existence of a left dualization for A (i.e., A is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case a Hopf module construction for A always exists. We recover from the general theory developed here results on coquasi-Hopf algebras.     

On corepresentations of multiplier Hopf algebras

We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.     

Finitely Cogenerated Distributive Modules are Q-Cocyclic

Abstract

In this paper we consider a generalization of Vámos's proposition which asserts that finitely generated artinian and distributive modules are cyclic. Also, we show its duality as an answer to the problem which is shown by him (See Vámos, [Vámos, P. (1978). Finitely generated artinian and distributive modules are cyclic. Bull. London Math. Soc. 10(3):287–288]).     

New coefficients for Hopf Cyclic Cohomology

In this note, the categories of coefficients for Hopf cyclic cohomology of comodule algebras and comodule coalgebras are extended. We show that these new categories have two proper different subcategories where the smallest one is the known category of stable anti Yetter–Drinfeld modules. We prove that components of Hopf cyclic cohomology such as cup products work well with these new coefficients.     

Rational Modules for Corings

Abstract

The so called dense pairings were studied mainly by Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with Gómez-Torricillas and Lobillo to the so called rational pairings over a commutative ground ring R to study the interplay between the comodules of an R-coalgebra C and the modules of an R-algebra A that admits an R-algebra morphism κ : A → C*. Such pairings, satisfying the so called α-condition, were called in the author's dissertation measuring α-pairings and can be considered as the corner stone in his study of duality theorems for Hopf algebras over commutative rings. In this paper we lay the basis of the theory of rational modules of corings extending results on rational modules for coalgebras to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g., Doi-Koppinen modules, alternative Doi-Koppinen modules) generalizing results of Doi, Koppinen, Menini et al.     

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.

     

Semisimplicity of the Categories of Yetter–Drinfeld Modules and Long Dimodules

Abstract

Let k be a field, and H a Hopf algebra with bijective antipode. If H is commutative, noetherian, semisimple and cosemisimple, then the category _H 𝒴𝒟 ^H of Yetter–Drinfeld modules is semisimple. We also prove a similar statement for the category of Long dimodules, without the assumption that H is commutative.     

Weak bimonoids in duoidal categories

Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg–Moore category of the induced monad. Making appropriate assumptions on the duoidal category in question, the fundamental theorem of Hopf modules is proven which says that the category of modules over one of the base monoids is equivalent to the category of Hopf modules if and only if a Galois-type comonad morphism is an isomorphism.