Characterization of quasi-Yetter–Drinfeld modules

In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra [Formula: see text], which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of [Formula: see text]-modules is equivalent to the category [Formula: see text] of quasi-Yetter–Drinfeld modules. Next, we prove that [Formula: see text] is equivalent to the category of generalized Hopf bimodules. Finally, we show that [Formula: see text] is also equivalent to the category of quasi-coactions over some Majid’s braided group if [Formula: see text] is quasi-triangular.

Gauge transformations for quasitriangular quasi-Turaev group coalgebras

The purpose of this paper is to introduce an equivalence relation on quasitriangular quasi-Turaev group coalgebras such that the categories of representations of two quasitriangular quasi-Turaev group coalgebras are tensor [Formula: see text]-equivalent. As an application, we construct a new Turaev braided group category by a gauge transformation and give a new solution of Yang–Baxter type equation.

TWISTED DRINFELD DOUBLES AND REPRESENTATIONS OF A HOPF ALGEBRA

In this paper, we consider a class of non-commutative and non-cocommutative Hopf algebras Hp(α, q, m) and then show that these Hopf algebras can be realized as a quantum double of certain Hopf algebras with respect to Hopf skew pairings (Ap(q, m), Bp(q, m), τα). Furthermore, using the Hopf skew pairing with appropriate group homomorphisms: ϕ : π → Aut (Ap(q, m)) and ψ : π → Aut (Bp(q, m)), we construct a twisted Drinfeld double D(Ap(q, m), Bp(q, m), τ; ϕ, ψ) which is a Turaev [Formula: see text]-coalgebra, where the group [Formula: see text] is a twisted semi-direct square of a group π. Then we obtain its quasi-triangular Turaev [Formula: see text]-coalgebra structure. We also study irreducible representations of Hp(1, q, m) and construct a corresponding R-matrix. Finally, we introduce the notion of a left Yetter–Drinfeld category over a Turaev group coalgebra and show that such a category is a Turaev braided group category by a direct proof, without center construction. As an application, we consider the case of the quasi-triangular Turaev [Formula: see text]-coalgebra structure on our twisted Drinfeld double.

3D Braided Geometry Structures Based on Space Group

3D braided group theory is dissertated. The analysis procedure is described from the existing braided geometry structure to the braided space group; 3D braided geometrical structures are finally described by means of group theory. Some of novel 3D braided structures are deduced from the braided space groups. By describing the 3D braided materials with braided space point and braided space groups, the braided space groups are not always the same as symmetry groups of crystallographic because novel lattices can be produced and the reflection operation cannot exist in braided space point groups. Braided point and space groups are theoretical basis for deriving the novel braided geometry structure.