Braided Hopf algebras have attracted much attention in both mathematics and mathematical physics (see e.g. [2][4][14][16][17][18][20][23]). The classification of finite dimensional Hopf algebras is interesting and important for their applications (see [1] [24]). Braided Hopf algebras play an important role in the classification of finite-dimensional pointed Hopf algebras (e.g. [1][2] [21]). The...

We provide an analog of the Joyal-Street center construction and of the Kassel-Turaev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of a YetterDrinfeld module over a crossed group coalgebra H and we prove that both the category of Yetter-Dri...

Let H be a crossed group-cograded Hopf quasigroup. We first introduce the notion of p-Yetter–Drinfeld quasimodule over H. If antipode is bijective, we show that category YDQ(H) Yetter–Drinfeld quasimodules category, and subcategory YD(H) modules braided category.

in this paper, we presented algebroids and crossed modules of algebroids. we also presented pullback crossed module of algebroids and we then define pushout crossed modules of algebroids.

We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2. Introduction Crossed modules defined by Whitehead, [23], are algebraic models of connected (weak homotopy) 2-types. Crossed squares as introduced by Loday and GuinWalery, [22], model connected 3-types. Crossed n-cubes model connected (n + 1)types, (cf. [21])....

Abstract We study noncommutative differential structures on permutation groups SN , defined by conjugacy classes. The 2-cycles class defines an exterior algebra ΛN which is a super analogue of the quadratic algebra EN for Schubert calculus on the cohomology of the flag variety. Noncommutative de Rahm cohomology and moduli of flat connections are computed for N < 6. We find that flat connections...

In the definition of a crossed module $(T,G,rho)$, the actions of the group $T$ and $G$ on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary crossed module. Therefore, we get the category ${bf GCM}$, of all generalized crossed modules and generalized crossed module morphisms between them, and investigate som...

‎We firstly prove the completeness of the category of crossed modules in a modified category of interest‎. ‎Afterwards‎, ‎we define pullback crossed modules and pullback cat objects that are both obtained by pullback diagrams with extra structures on certain arrows‎. ‎These constructions unify many corresponding results for the cases of groups‎, ‎commutative algebras and can also be adapted to ...