We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products $H_4\bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4,H_4,▹,◃)$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for...

We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space $V$. The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on $V$ (resp. graded Loday structures on $V$, sequences that we call Loday infinity structures on $V$). We prove a minimal model theorem for Loday infinity algebras and observe that the ${\text{Lod}}_{\infty }$ category contains the ${\text{L}}_{\infty }$ category as...

We construct quantum commutators on comodule algebras over coquasitriangular Hopf algebras, so that they are quantum group coinvariant and have the generalized antisymmetry and Leibniz properties. If the coquasitriangular Hopf algebra is additionally cotriangular, then the quantum commutators satisfy a generalized Jacobi identity, and turn the comodule algebra into a quantum Lie algebra. Moreover, we investigate the projective and injective dimensions of some Doi-Hopf modules over a quantum commutative...

We define polynomial $H$-identities for comodule algebras over a Hopf algebra $H$ and establish general properties for the corresponding $T$-ideals. In the case $H$ is a Taft algebra or the Hopf algebra $E\left(n\right)$, we exhibit a finite set of polynomial $H$-identities which distinguish the Galois objects over $H$ up to isomorphism.

We introduce the notions of silting comodules and finitely silting comodules in quasi-finite category, and study some properties of them. We investigate the torsion pair and dualities which are related to finitely silting comodules, and give the equivalences among silting comodules, finitely silting comodules, tilting comodules and finitely tilting comodules.

In this paper, for a cocommutative Hopf algebra H in a symmetric closed category C with basic object K, we get an isomorphism between the group of isomorphism classes of Galois H-objects with a normal basis and the second cohomology group H2(H,K) of H with coefficients in K. Using this result, we obtain a direct sum decomposition for the Brauer group of H-module Azumaya monoids with inner action:BMinn(C,H) ≅ B(C) ⊕ H2(H,K)In particular, if C is the symmetric closed category of C-modules with K a...

Nous considérons deux algèbres de Hopf graduées connexes en interaction, l’une étant un comodule-cogèbre sur l’autre. Nous montrons comment définir l’analogue du groupe de renormalisation et de la fonction Bêta de Connes-Kreimer lorsque la bidérivation de graduation est remplacée par une bidérivation provenant d’un caractère infinitésimal de la deuxième algèbre de Hopf.

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...

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.

We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product, and consider some examples. Furthermore, we introduce the notion of a generalized partial twisted smash product and discuss a necessary condition under which such partial smash product forms a Hopf algebra. Based on these notions and properties, we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra.

Let $\pi$ be a group, and $H$ be a semi-Hopf $\pi$-algebra. We first show that the category ${}_Hℳ$ of left $\pi$-modules over $H$ is a monoidal category with a suitably defined tensor product and each element $\alpha$ in $\pi$ induces a strict monoidal functor $F_{\alpha }$ from ${}_Hℳ$ to itself. Then we introduce the concept of quasitriangular semi-Hopf $\pi$-algebra, and show that a semi-Hopf $\pi$-algebra $H$ is quasitriangular if and only if the category ${}_Hℳ$ is a braided monoidal category and $F_{\alpha }$ is a strict braided monoidal functor for any $\alpha \in \pi$. Finally,...

We define and investigate Ringel-Hall algebras of coalgebras (usually infinite-dimensional). We extend Ringel's results [Banach Center Publ. 26 (1990) and Adv. Math. 84 (1990)] from finite-dimensional algebras to infinite-dimensional coalgebras.