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...

Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products $(A{♮}_{R}B,\alpha \otimes \beta )$. Moreover, necessary and sufficient conditions for $(A{♮}_{R}B,\alpha \otimes \beta )$ to be a cobraided Hom-Hopf algebra are given.

Soit $A_1\left(ℂ\right)$ la première algèbre de Weyl sur $ℂ$. La codimension B-W d’un idéal à droite non nul $I$ de $A_1\left(ℂ\right)$ a été introduite par Yuri Berest et George Wilson. Nous montrons d’une part que cette codimension est invariante par la relation de Stafford : si $x\in Q_1=\mathrm{Frac}\left(A_1\left(ℂ\right)\right)$, le corps de fractions de $A_1\left(ℂ\right)$, et si $\sigma \in \mathrm{Aut}\left(A_1\left(ℂ\right)\right)$, le groupe des $ℂ$-automorphismes de $A_1\left(ℂ\right)$, sont tels que $J=x\sigma \left(I\right)$ soit un idéal à droite de $A_1\left(ℂ\right)$, alors $\mathrm{codim}I=\mathrm{codim}x\sigma \left(I\right)$. Nous relions d’autre part la codimension d’un idéal $I$ à la codimension de Gail Letzter-Makar Limanov, de $\mathrm{End}\left(I\right)$, l’anneau des endomorphismes...

We give axiomatic conditions in order to calculate the local cohomology of some idempotent kernel functors. These results lie in some new dimension introduced by T. Levasseur for Auslander-Gorenstein rings. Under some hypothesis, we generalize previous results.

Let $G$ be a finite group with a normal subgroup $N$ such that $C_G\left(N\right)\le N$. It is shown that under some conditions, Coleman automorphisms of $G$ are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.

A new class of abelian $p$-groups with all high subgroups isomorphic is defined. Commutative modular and semisimple group algebras over such groups are examined. The results obtained continue our recent statements published in Comment. Math. Univ. Carolinae (2002).

Let $G$ be a $p$-mixed abelian group and $R$ is a commutative perfect integral domain of $charR=p>0$. Then, the first main result is that the group of all normalized invertible elements $V\left(RG\right)$ is a $\Sigma$-group if and only if $G$ is a $\Sigma$-group. In particular, the second central result is that if $G$ is a $\Sigma$-group, the $R$-algebras isomorphism $RA\cong RG$ between the group algebras $RA$ and $RG$ for an arbitrary but fixed group $A$ implies $A$ is a $p$-mixed abelian $\Sigma$-group and even more that the high subgroups of $A$ and $G$ are isomorphic, namely, ${\mathscr{H}}_A\cong {\mathscr{H}}_G$. Besides,...