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We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced \cite{AGV1} bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of $A$, called $\Omega$-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study $V$-universal measuring coalgebras and $V$-universal comeasuring algebras between $\Omega$-algebras $A$ and $B$, relative to a fixed subspace $V$ of $\Vect(A,B)$. By considering the case $A=B$, we derive the notion of a $V$-universal (co)acting bialgebra (and Hopf algebra) for a given algebra $A$. In particular, this leads to a refinement of the existence conditions for the Manin--Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the $V$-universal acting bi/Hopf algebra and the finite dual of the $V$-universal coacting bi/Hopf algebra under certain conditions on $V$ in terms of the finite topology on $\End_F(A)$., Comment: To appear in Commun. Contemp. Math

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.

Given a subgroup H of a finite group G, we begin a systematic study of the partial representations of G that restrict to global representations of H. After adapting several results from [DEP00] (which correspond to the case where H is trivial), we develop further an effective theory that allows explicit computations. As a case study, we apply our theory to the symmetric group and its subgroup of permutations fixing 1: this provides a natural extension of the classical representation theory of the symmetric group., 31 pages

We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal. Dually, we consider $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal categories with compatible $\mathsf{Lax}^+\mathsf{Oplax}^0$- and $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal structures. We introduce comonoids in $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal categories, monoids in $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories and bimonoids in $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$- duoidal categories. Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts. This assigns a $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal category $\mathsf D$ to any symmetric monoidal category $\mathsf V$. The unital $\mathsf{BiHom}$-monoids, counital $\mathsf{BiHom}$-comonoids, and unital and counital $\mathsf{BiHom}$-bimonoids in $\mathsf V$ are identified with the monoids, comonoids and bimonoids in $\mathsf D$, respectively., 40 pages, a few figures of commutative diagrams

We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor–Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context., SCOPUS: ar.j, info:eu-repo/semantics/published

We introduce the notion of a dilation for a partial representation (that is, a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (that is, a partial module algebra) coincides with the enveloping action (or globalization). This construction leads to categorical equivalences between the category of partial H-modules, a category of (global) H-modules endowed with a projection satisfying a suitable commutation relation and the category of modules over a (global) smash product constructed upon H, from which we deduce the structure of a Hopfish algebra on this smash product. These equivalences are used to study the interactions between partial and global representation theory., SCOPUS: ar.j, info:eu-repo/semantics/published

We describe the split extension classifiers in the semi-abelian category of cocommutative Hopf algebras over an algebraically closed field of characteristic zero. The categorical notions of centralizer and of center in the category of cocommutative Hopf algebras is then explored. We show that the categorical notion of center coincides with the one that is considered in the theory of general Hopf algebras., 24 pages

The purpose of this article is to prove that the category of cocommutative Hopf $K$-algebras, over a field $K$ of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie $K$-algebras, respectively.

The duality between partial actions (partial $H$-module algebras) and co-actions (partial $H$-comodule algebras) of a Hopf algebra $H$ is fully explored in this work. A connection between partial (co)actions and Hopf algebroids is established under certain commutativity conditions. Moreover, we continue this duality study, introducing also partial $H$-module coalgebras and their associated $C$-rings, partial $H$-comodule coalgebras and their associated cosmash coproducts, as well as the mutual interrelations between these structures., v3: strongly revised version

We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf algebra H, there is a natural isomorphism of Lie algebras Q(H) * ≅ P(H {ring operator}), where Q(H) * is the dual Lie algebra of the Lie coalgebra of indecomposables of H, and P(H {ring operator}) is the Lie algebra of primitive elements of the Sweedler dual of H. We apply our theory to Turaev's Hopf group-(co)algebras. © 2014 Elsevier Inc., SCOPUS: ar.j, info:eu-repo/semantics/published

In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra $H$, one can associate it to a Hopf algebroid $H_{par}$ which has the universal property that each partial representation of $H$ can be factorized by an algebra morphism from $H_{par}$. We define also the category of partial modules over a Hopf algebra $H$, which is the category of modules over its associated Hopf algebroid $H_{par}$. The Hopf algebroid structure of $H_{par}$ enables us to enhance the category of partial $H$ modules with a monoidal structure and such that the algebra objects in this category are the usual partial actions. Some examples of categories of partial $H$ modules are explored. In particular we can describe fully the category of partially $\mathbb{Z}_2$-graded modules., 39 pages, version 2: some additions in the section on partial gradings and minor corrections elsewhere

We introduce the category of bicomodules for a comonad on a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups using cointegrations into bicomodules. We present two applications: the characterization of coseparable corings stated in [14], and the characterization of coseparable coalgebra coextensions stated in [19].

We compare several quasi-Frobenius-type properties for corings that appeared recently in literature and provide several new characterizations for each of these properties. By applying the theory of Galois comodules with a firm coinvariant ring, we can characterize a locally quasi-Frobenius (quasi-co-Frobenius) coring as a locally projective generator in its category of comodules.

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule $\Sigma$ of an $A$-coring $\cC$. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring $\cC$ or the comodule $\Sigma$ is finitely generated and projective as an $A$-module. That is, we obtain relations between the category of $\cC$-comodules and the category of firm modules for a firm ring $R$, which is an ideal of the endomorphism algebra $^\cC(\Sigma)$. For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map., Comment: LaTeX, 35 pages. v2: Minor changes including the title, examples added in Section 2

We construct comatrix corings on bimodules without finiteness conditions by using firm rings. This leads to the formulion of a notion of Galois coring which plays a key role in the statement of a Noncommutative Faithfully Flat Descent for comodules which generalizes previous versions. In particular, infinite comatrix corings fit in our general theory.

We show the close connection between appearingly different Galois theories for comodules introduced recently in [J. G\'omez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, arXiv:math.RA/0509106.] and [R. Wisbauer, On Galois comodules (2004), to appear in Comm. Algebra.]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules., Comment: 19 pages, some corrections and additions to version 1

We propose a class of infinite comatrix corings, and describe them as colimits of systems of usual comatrix corings. The infinite comatrix corings of El Kaoutit and G\'omez Torrecillas are special cases of our construction, which in turn can be considered as a special case of the comatrix corings introduced recently by G\'omez Torrecillas an the third author., Comment: 23 pages, some corrections and additions to the previous version, and a renumbering of theorems in section 4

We unify and generalize different notions of local units and local projectivity. We investigate the connection between these properties by constructing elementary algebras from locally projective modules. Dual versions of these constructions are discussed, leading to corings with local comultiplications, corings with local counits and rings with local multiplications., 22 pages, including a correction to Proposition 1.1

Using the theory of corings, we generalize and unify Morita contexts introduced by Chase and Sweedler, Doi, and Cohen, Fischman and Montgomery. We discuss when the contexts are strict. We apply our theory corings arising from entwining structures, and this leads us to the notion of cleft entwining structure., 27 pages

We study co-Frobenius and more generally quasi-co-Frobenius corings over arbitrary base rings and over PF base rings in particular. We generalize some results about co-Frobenius and quasi-co-Frobenius coalgebras to the case of non-commutative base rings and give several new characterizations for co-Frobenius and more generally quasi-co-Frobenius corings, some of them are new even in the coalgebra situation. We construct Morita contexts to study Frobenius properties of corings and a second kind of Morita contexts to study adjoint pairs. Comparing both Morita contexts, we obtain our main result that characterizes quasi-co-Frobenius corings in terms of a pair of adjoint functors (F,G) such that (G,F) is locally quasi-adjoint in a sense defined in this note.