The Monoidal Category of Yetter-Drinfeld Modules over a Weak Braided Hopf Algebra

In this paper we introduce the notion of weak operators and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non-strict monoidal category. It is also shown that this category is not trivial, that is to say, it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra.

A method of constructing braided Hopf algebras

Let A and B be two Hopf algebras and R ( Hom(B ( A, A ( B), the twisted tensor product Hopf algebra A#RB was introduced by S. Caenepeel et al in [3] and further studied in our recent work [6]. In this paper we give the necessary and sufficient conditions for A#RB to be a Hopf algebra with a projection. Furthermore, a braided Hopf algebra A is constructed by twisting the multiplication of A through a (?, R)-pair (A, B). Finally we give a method to construct Radford's biproduct directly by defining the module action and comodule action from the twisted tensor biproduct. 2010 Mathematics Subject Classifications. 16W30. .

Making nontrivially associated modular categories from finite groups

We show that the double𝒟of the nontrivially associated tensor category constructed from left coset representatives of a subgroup of a finite groupXis a modular category. Also we give a definition of the character of an object in this category as an element of a braided Hopf algebra in the category. This definition is shown to be adjoint invariant and multiplicative on tensor products. A detailed example is given. Finally, we show an equivalence of categories between the nontrivially associated double𝒟and the trivially associated category of representations of the Drinfeld double of the groupD(X).

DIFFERENTIAL CALCULUS ON INHOMOGENEOUS QUANTUM GROUPS

We investigate the question of covariant differential calculi on the bosonisation of a coquasitriangular Hopf algebra and an associated braided Hopf algebra. As a result we present a general way of obtaining such calculi on inhomogeneous quantum groups.

SOLUTIONS OF THE YANG-BAXTER EQUATIONS FROM BRAIDED-LIE ALGEBRAS AND BRAIDED GROUPS

We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring k[x] of polynomials in one variable, regarded as a braided-line. Representations of the extended Artin braid group for braids in the complement of S1 are also obtained by the same method.