Factorizable quasi-Hopf algebras. Applications

We define the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the Drinfeld double $D(H)$ of any finite dimensional quasi-Hopf algebra $H$ is factorizable, and we characterize $D(H)$ when $H$ itself is factorizable. Finally, we prove that any finite dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we obtain that the Drinfeld double $D(H)$ is a unimodular quasi-Hopf algebra.Comment: 35 page

Quiver Bialgebras and Monoidal Categories

We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories.Comment: 10 page

Relative exact covers

summary:Recently Rim and Teply [11] found a necessary condition for the existence of $\sigma$-torsionfree covers with respect to a given hereditary torsion theory for the category $R$-mod. This condition uses the class of $\sigma$-exact modules; i.e. the $\sigma$-torsionfree modules for which every its $\sigma$-torsionfree homomorphic image is $\sigma$-injective. In this note we shall show that the existence of $\sigma$-torsionfree covers implies the existence of $\sigma$-exact covers, and we shall investigate some sufficient conditions for the converse

Precovers

summary:Let $\mathcal G$ be an abstract class (closed under isomorpic copies) of left $R$-modules. In the first part of the paper some sufficient conditions under which $\mathcal G$ is a precover class are given. The next section studies the $\mathcal G$-precovers which are $\mathcal G$-covers. In the final part the results obtained are applied to the hereditary torsion theories on the category on left $R$-modules. Especially, several sufficient conditions for the existence of $\sigma$-torsionfree and $\sigma$-torsionfree $\sigma$-injective covers are presented