Related papers: Duality Theorem and Drinfeld Double in Braided Ten…

Let $H$ be an infinite-dimensional braided Hopf algebra and assume that the braiding is symmetric on $H$ and its quasi-dual $H^d$. We prove the Blattner-Montgomery duality theorem, namely we prove $$ (R # H)# H^{d} \cong R \otimes (H #…

Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$, Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd…

We classify equivalence classes of Hopf algebra quotient pairs $(D,\theta)$ of the Drinfeld double $D(G)$ of a finite group scheme $G$ over an algebraically closed field $\mathbf{k}$ of characteristic $p\ge 0$, in terms of group…

Drinfeld showed that any finite dimensional Hopf algebra \G extends to a quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the construction of a so--called diagonal crossed product developed by the authors, we…

Let $(A,\alpha)$ be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel'd double $D(A)=(A^{op}\bowtie A^{\ast},\alpha\otimes(\alpha^{-1})^{\ast})$ in the setting of Hom-Hopf algebras by two ways, one of…

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group…

In this paper, the Drinfeld center of a monoidal category is generalized to a class of mixed Drinfeld centers. This gives a unified picture for the Drinfeld center and a natural Heisenberg analogue. Further, there is an action of the former…

The Drinfeld double associated to the weak multiplier Hopf ($*$-) algebra pairing $\left\langle A, B\right\rangle$ is constructed. We show that the Drinfeld double is again a weak multiplier Hopf ($*$-) algebra. If $A$ and $B$ are algebraic…

A certain class of rank two pointed Hopf algebras is considered. The simple modules of their Drinfel'd double is described using Radford's method \cite{rad}. The socle of the tensor product of two such modules is computed and a formula…

Let $(H, \beta)$ be a monoidal Hom-Hopf algebra with the bijective antipode $S$, In this paper, we mainly construct the Drinfel'd codouble $T(H)=(H^{op}\otimes H^{*}, \beta\otimes \beta^{*-1})$ and $\widehat{T(H)}=( H^{*}\otimes H^{op},…

For a Hopf algebra B with bijective antipode, we show that the Heisenberg double H(B^*) is a braided commutative Yetter--Drinfeld module algebra over the Drinfeld double D(B). The braiding structure allows generalizing H(B^*) =…

Let A be a Hopf algebra in a braided rigid category B. In the case B admits a coend C, which is a Hopf algebra in B, we defined in 2008 the double D(A) of A, which is a quasitriangular Hopf algebra in B whose category of modules is…

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 $H$…

Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH).…

Blatter-Montgomery duality theorem is generalized into braided tensor categories. It is shown that $Hom(V,W)$ is a braided Yetter-Drinfeld module for any two braided Yetter-Drinfeld modules $V$ and $W$.

Let $(H, \sigma)$ be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras,…

We study the Yetter--Drinfeld D(B)-module algebra structure on the Heisenberg double H(B^*) endowed with a "heterotic" action of the Drinfeld double D(B). This action can be interpreted in the spirit of Lu's description of H(B^*) as a twist…

Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful…

We briefly report on our result that the braided tensor product algebra of two module algebras $A_1,A_2$ of a quasitriangular Hopf algebra $H$ is equal to the ordinary tensor product algebra of $H_1$ with a subalgebra isomorphic to $A_2$…

We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors ?^{**} and D tensor ^{**}? tensor D^{-1}. This…