Hopf-Galois extensions for monoidal Hom-Hopf algebras are investigated. As the main result, Schneider's affineness theorem in the case of monoidal Hom-Hopf algebras is shown in terms of total integrals and Hopf-Galois extensions. In addition, we obtain an affineness criterion for relative Hom-Hopf modules which is associated with faithfully flat Hopf-Galois extensions of monoidal Hom-Hopf algebras.

We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if π:A → Mₙ(ℂ) is a finite-dimensional representation of a Hopf C*-algebra, we prove that the idempotent state associated to its Hopf image A' must be the convolution Cesàro limit of the linear functional φ = tr ∘ π. We then discuss some consequences of this result, notably to inner linearity questions.

The incidence coalgebras $K^{\square }I$ of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form $q^{•}:{\mathbb{Z}}^{\left(I\right)}\to \mathbb{Z}$, where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category $K^{\square }I-comod$ of finite-dimensional left $K^{\square }I$-modules is equivalent to the tameness of the category $K^{\square }I-Como{d}_{fc}$ of finitely copresented left $K^{\square }I$-modules. Hence, the tame-wild dichotomy for the coalgebras $K^{\square }I$ is deduced. Moreover, we prove that for an interval finite ̃ *ₘ-free...

The half-liberated orthogonal group $O_n^{*}$ appears as intermediate quantum group between the orthogonal group $O_n$, and its free version $O_n^{+}$. We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between $O_n^{*}$ and $U_n$, a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that...

A description of the commutator of a normal subcategory of the fusion category of representation Rep A of a semisimple Hopf algebra A is given. Formulae for the kernels of representations of Drinfeld doubles D(G) of finite groups G are presented. It is shown that all these kernels are normal Hopf subalgebras.

We introduce the notion of a lazy 2-cocycle over a monoidal Hom-Hopf algebra and determine all lazy 2-cocycles for a class of monoidal Hom-Hopf algebras. We also study the extension of lazy 2-cocycles to a Radford Hom-biproduct.

On étudie dans cet article les notions d’algèbre à homotopie près pour une structure définie par deux opérations $\text{.}$ et $[,]$. Ayant déterminé la structure des $G_{\infty }$ algèbres et des $P_{\infty }$ algèbres, on généralise cette construction et on définit la stucture des $(a,b)$-algèbres à homotopie près. Etant donnée une structure d’algèbre commutative et de Lie différentielle graduée pour deux décalages des degrés donnés par $a$ et $b$, on donnera une construction explicite de l’algèbre à homotopie près associée et on précisera...

Tilting theory plays an important role in the representation theory of coalgebras. This paper seeks how to apply the theory of localization and colocalization to tilting torsion theory in the category of comodules. In order to better understand the process, we give the (co)localization for morphisms, (pre)covers and special precovers. For that reason, we investigate the (co)localization in tilting torsion theory for coalgebras.

We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, φ: C → D is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras C and D coincide if and only if ${\sum }_{i\in I}\epsilon \left(a^i\right)b^i={\sum }_{i\in I}{a}^i\epsilon \left(b^i\right)$ for all ${\sum }_{i\in I}{a}^i\otimes b^i\in C{◻}_{D}C$. In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.

The so-called “invariance under twisting” for twisted tensor products of algebras is a result stating that, if we start with a twisted tensor product, under certain circumstances we can “deform” the twisting map and we obtain a new twisted tensor product, isomorphic to the given one. It was proved before that a number of independent and previously unrelated results from Hopf algebra theory are particular cases of this theorem. In this article we show that some more results from literature are particular...

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite...