We prove a structure theorem for Yetter-Drinfel'd Hopf algebras over groups of prime order that are nontrivial, cocommutative, and cosemisimple: Under certain assumptions on the base field, these algebras can be decomposed into a tensor product of the dual group ring of the group of prime order and an ordinary group ring of some other group. This tensor product is a crossed product as an algebra and an ordinary tensor product as a coalgebra. In particular, the dimension of such a Yetter-Drinfel'd Hopf algebra is divisible by the prime under consideration. We also find explicit examples of such Yetter-Drinfel'd Hopf algebras and apply the result to the classification program for semisimple Hopf algebras.
Under suitable assumptions on the base field, we prove that a commutative semisimple Yetter-Drinfel'd Hopf algebra over a finite abelian group is trivial, i.e., is an ordinary Hopf algebra, if its dimension is relatively prime to the order of the finite abelian group. Furthermore, we prove that a finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra contains a trivial Yetter-Drinfel'd Hopf subalgebra of dimension greater than one, at least if the Yetter-Drinfel'd Hopf algebra itself has dimension greater than one.
By constructing explicit examples, we show that the core of a group-like element in a cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of a finite abelian group is not always completely trivial.
We introduce Yetter-Drinfeld modules over a weak Hopf algebra $H$, and show that the category of Yetter-Drinfeld modules is isomorphic to the center of the category of $H$-modules. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If $H$ is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show …
We define and study a class of entwined modules (stable anti-Yetter-Drinfeld modules) that serve as coefficients for the Hopf-cyclic homology and cohomology. In particular, we explain their relationship with Yetter-Drinfeld modules and Drinfeld doubles. Among sources of examples of stable anti-Yetter-Drinfeld modules, we find Hopf-Galois extensions with a flipped version of the Miyashita-Ulbrich action.
In the present work, we study Yetter-Drinfeld algebras over a pairing of multiplier Hopf algebras. Our main motivation is the construction of a self-dual theory of (C*-)algebraic quantum transformation groupoids. Instead of the standard characterization of Yetter-Drinfeld algebras given in the case of Hopf algebras, we develop an equivalent "only coaction" characterization in the framework of multiplier Hopf algebras. Finally, as a special case, we focus on Yetter-Drinfeld structures over Van Daele's algebraic quantum groups.
In this work, we study another characterization of Yetter-Drinfeld algebras over finite-dimensional Hopf algebras. We show the equivalence between this characterization, called the "only coaction" characterization, and the standard "action-coaction" characterization. This modern approach for Yetter-Drinfeld algebras is one of the key ingredient in a self-dual theory of quantum transformation groupoids arising from actions of quantum groups.
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 double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd $\mathscr{D}$-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.
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 of D(B). In terms of the braiding of Yetter--Drinfeld modules, H(B^*) is braided commutative. By the Brzezinski--Militaru theorem, H(B^*)#D(B) is then a Hopf algebroid over H(B^*). For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_q(s…
Let $k$ be a field, and $H$ a Hopf algebra with bijective antipode. If $H$ is commutative, noetherian, semisimple and cosemisimple, then the category ${}_{H}{\mathcal {YD}}^H$ of Yetter-Drinfeld modules is semisimple. We also prove a similar statement for the category of Long dimodules, without the assumption that $H$ is commutative.
We prove that a finite-dimensional Hopf algebra with the dual Chevalley Property over a field of characteristic zero is quasi-isomorphic to a Radford-Majid bosonization whenever the third Hochschild cohomology group in the category of Yetter-Drinfeld modules of its diagram with coefficients in the base field vanishes. Moreover we show that this vanishing occurs in meaningful examples where the diagram is a Nichols algebra.
For a quasi-Hopf algebra $H$, a left $H$-comodule algebra $\mf{B}$ and a right $H$-module coalgebra $C$ we will characterize the category of Doi-Hopf modules ${}^C{\cal M}(H)_{\mf{B}}$ in terms of modules. We will also show that for an $H$-bicomodule algebra $\mb{A}$ and an $H$-bimodule coalgebra $C$ the category of generalized Yetter-Drinfeld modules ${}_{\mb{A}}{\cal YD}(H)^C$ is isomorphic to a certain category of Doi-Hopf modules. Using this isomorphism we will transport the properties from the category of Doi-Hopf modules to the category of generalized Yetter-Drinfeld modules.
We introduce and study a general concept of integral of a threetuple (H, A, C), where H is a Hopf algebra acting on a coalgebra C and coacting on an algebra A. In particular, quantum integrals associated to Yetter-Drinfel'd modules are defined. Let A be an H-bicomodule algebra, $^H {\cal YD}_A$ be the category of (generalized) Yetter-Drinfel'd modules and $B$ the subalgebra of coinvariants of the Verma structure of $A$. We introduce the concept of quantum Galois extensions and we prove the affineness criterion in a quantum version.
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^*) = B^{*cop}\braid B to "Heisenberg n-tuples" and "chains" ...\braid B^{*cop}\braid B \braid B^{*cop}\braid B\braid..., all of which are Yetter--Drinfeld D(B)-module algebras. For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_qsl(2).
Over fields of arbitrary characteristic we classify all braid-indecomposable tuples of at least two absolutely simple Yetter-Drinfeld modules over non-abelian groups such that the group is generated by the support of the tuple and the Nichols algebra of the tuple is finite-dimensional. Such tuples are classified in terms of analogs of Dynkin diagrams which encode much information about the Yetter-Drinfeld modules. We also compute the dimensions of these finite-dimensional Nichols algebras. Our proof uses the Weyl groupoid of a tuple of simple Yetter-Drinfeld modules.
Let $H$ be a crossed group-cograded Hopf quasigroup. We first introduce the notion of $p$-Yetter-Drinfeld quasimodule over $H$. If the antipode of $H$ is bijective, we show that the category $\mathscr Y\mathscr D\mathscr Q(H)$ of Yetter-Drinfeld quasimodules over $H$ is a crossed category, and the subcategory $\mathscr Y\mathscr D(H)$ of Yetter-Drinfeld modules is a braided crossed category.
Let $H$ be a dual quasi-Hopf algebra. In this paper we will firstly introduce all possible categories of Yetter-Drinfeld modules over $H$, and give explicitly the monoidal and braided structure of them. Then we prove that the category $^H_H\mathcal{YD}^{fd}$ of finite-dimensional left-left Yetter-Drinfeld modules is rigid. Finally we will study the braided cocommunitivity of $H_0$ in $^H_H\mathcal{YD}$.
For a regular multiplier Hopf algebra $A$, the Yetter-Drinfel'd module category ${}_{A}\mathcal{YD}^{A}$ is equivalent to the centre $Z({}_{A}\mathcal{M})$ of the unital left $A$-module category ${}_{A}\mathcal{M}$. Then we introduce the generalized $(α, β)$-Yetter-Drinfel'd module categories ${}_{A}\mathcal{GYD}^{A}(α, β)$, which are treated as components of a braided $T$-category. Especially when $A$ is a coFrobenius Hopf algebra, ${}_{A}\mathcal{YD}^{A}(α, β)$ is isomorphic to the unital $\hat{A} \bowtie A(α, β)$-module category ${}_{\hat{A} \bowtie A(α, β)}\mathcal{M}$. Finally for a Yette…
Let $π$ be a group. The aim of this paper is to construct the category of Yetter-Drinfeld modules over the quasi-Turaev group coalgebra $H=(\{H_\a\}_{\a\inπ},Δ,\varepsilon,S,Φ)$, and prove that this category is isomorphic to the center of the representation category of $H$. Therefore a new Turaev braided group category is constructed.
The aim of this paper is to define and study Yetter-Drinfeld modules over weak Hom-Hopf algebras. We show that the category ${}_H{\cal WYD}^H$ of Yetter-Drinfeld modules with bijective structure maps over weak Hom-Hopf algebras is a rigid category and a braided monoidal category, and obtain a new solution of quantum Hom-Yang-Baxter equation. It turns out that, If $H$ is quasitriangular (respectively, coquasitriangular)weak Hom-Hopf algebras, the category of modules (respectively, comodules) with bijective structure maps over $H$ is a braided monoidal subcategory of the category ${}_H{\cal WYD}…
Let $H$ be a Hopf algebra and $\mathcal{LR}(H)$ the category of Yetter-Drinfel'd-Long bimodules over $H$. We first give sufficient and necessary conditions for $\mathcal{LR}(H)$ to be symmetry and pseudosymmetry, respectively. We then introduce the definition of $u$-condition in $\mathcal{LR}(H)$ and discuss the relation between the $u$-condition and the symmetry of $\mathcal{LR}(H)$. Finally, we show that $\mathcal{LR}(H)$ over a triangular (cotriangular, resp.) Hopf algebra contains a rich symmetric subcategory.
In continuation of our recent work about smash product Hom-Hopf algebras in \cite{MLY}, we introduce Hom-Yetter-Drinfeld category $_H^H{\mathbb{YD}}$ via Radford biproduct Hom-Hopf algebra, and prove that the Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and $_H^H{\mathbb{YD}}$ is a pre-braided tensor category, where $(H, \b, S)$ is a Hom-Hopf algebra. Furthermore, we obtain that $(A^{\natural}_{\diamond} H,\aø\b)$ is a Radford biproduct Hom-Hopf algebra if and only if $(A,\a)$ is a Hopf algebra in the category $_H^H{\mathbb{YD}}$. At last, some examples and…
Let $H_1$ and $H_2$ be Hopf algebras which are not necessarily finite dimensional and $α,β\in Aut_{Hopf}(H_1), γ,δ\in Aut_{Hopf}(H_2)$. In this paper, we introduce a category ${}_{H_1}\mathcal{LR}_{H_2}(α, β, γ, δ)$, generalizing Yetter-Drinfeld-Long bimodules and construct a braided $T$-category $\mathcal{LR}(H_1,H_2)$ containing all the categories $_{H_1}\mathcal{LR}_{H_2}(α, β, γ, δ)$ as components. We also prove that if $(α, β, γ, δ)$ admits a quadruple in involution, then ${}_{H_1}\mathcal{LR}_{H_2}(α, β, γ, δ)$ is isomorphic to the usual category ${}_{H_1}\mathcal{LR}_{H_2}$ of Yetter-Dr…
Let $\left( H,R\right) $ be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field $k$. In this paper, we give the structure of irreducible objects of the Yetter-Drinfeld module category ${} {}_{H}^{H}\mathcal{YD}.$ Let $H_{R}$ be the Majid's transmuted braided group of $\left( H,R\right) ,$ we show that $H_{R}$ is cosemisimple. As a coalgebra, let $H_{R}=D_{1}\oplus\cdots\oplus D_{r}$ be the sum of minimal $H$-adjoint-stable subcoalgebras. For each $i$ $\left( 1\leq i\leq r\right) $, we choose a minimal left coideal $W_{i}$ of $D_{i}$, and we can define th…
In this paper, we generalizing the main result in Liu[10] to weak Hopf coquasigroups case. We first define and study group-cograded weak Hopf quasigroups, which generalize both group-cograded Hopf quasigroups and weak Hopf group-coalgebras. Then we introduce the notion of p-Yetter-Drinfeld weak quasimodule over group-cograded weak Hopf quasigroups H. If the antipode of H is bijective, we show that the category YDWQ(H) of Yetter-Drinfeld weak quasimodules over H is a crossed category, and the subcategory YD(H) of Yetter-Drinfeld modules is a braided crossed category.
We introduce the Hom-analogue of the L-R-smash product and use it to define the Hom-analogue of the diagonal crossed product. When H is a finite dimensional Hom-Hopf algebra with bijective antipode and bijective structure map, we define the Drinfeld double of H; its algebra structure is a Hom-diagonal crossed product and it has all expected properties, namely it is quasitriangular and modules over it coincide with left-right Yetter-Drinfeld modules over H.
We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces…
We study some problems related to lazy 2-cocycles, such as: extension of (lazy) 2-cocycles to a Drinfeld double and to a Radford biproduct, Yetter-Drinfeld data obtained from lazy 2-cocycles, lifting of projective representations afforded by lazy 2-cocycles.
We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with the bimodule category centres of the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.
We study Doi-Hopf data and Doi-Hopf modules for Hopf group-coalgebras. We introduce modules graded by a discrete Doi-Hopf datum; to a Doi-Hopf datum over a Hopf group coalgebra, we associate an algebra graded by the underlying discrete Doi-Hopf datum, using a smash product type construction. The category of Doi-Hopf modules is then isomorphic to the category of graded modules over this algebra. This is applied to the category of Yetter-Drinfeld modules over a Hopf group coalgebra, leading to the construction of the Drinfeld double. It is shown that this Drinfeld double is a quasitriangular gra…
We classify all groups G and all pairs (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the support of the direct sum of V and W generates G, the square of the braiding between V and W is not the identity, and the Nichols algebra of the direct sum of V and W admits a finite root system. As a byproduct, we determine the dimensions of such Nichols algebras, and several new families of finite-dimensional Nichols algebras are obtained. Our main tool is the Weyl groupoid of pairs of absolutely simple Yetter-Drinfeld modules over groups.
We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the a…
In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Be…
This paper generalizes the Drinfel'd twist to the multiplier Hopf algebra case. For a multiplier Hopf algebra $A$ with a twist $J$, we construct a new multiplier Hopf algebra $A^{J}$. Furthermore, if $A$ is quasitriangular, then $A^{J}$ is also. Finally, for a counimodular algebraic quantum group $A$, $A^{J}$ is an algebraic quantum group.
We investigate color Lie rings over finite group algebras and their universal enveloping algebras. We exhibit these universal enveloping algebras as PBW deformations of skew group algebras: Every color Lie ring over a finite group algebra with a particular Yetter-Drinfeld structure has universal enveloping algebra that is a quantum Drinfeld orbifold algebra. Conversely, every quantum Drinfeld orbifold algebra of a particular type arising from the action of an abelian group is the universal enveloping algebra of some color Lie ring over the group algebra. One consequence is that these quantum D…
We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier's version of the Fundamental Theorem for Hopf algebras. We introduce Yetter-Drinfeld modules over Hopf group-coalgebras using the center construction.
In this paper we mainly construct bicrossproduct for finite-dimensional monoidal Hom-Hopf algebra $(H,α)$, generalizing the Majid's bicrossproduct. Naturally the Hom-type bicrossproduct leads to Drinfel'd double $(H^{op}\bowtie H^{\ast}, α\otimes (α^{-1})^\ast)$ with a quasitriangular structure $R$ satisfying the quantum Hom-Yang-Baxter equations.
Let $(A,α)$ be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel'd double $D(A)=(A^{op}\bowtie A^{\ast},α\otimes(α^{-1})^{\ast})$ in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid's bicrossproduct for Hopf algebras (see \cite{M2}) and another one is to introduce the notion of dual pairs of of Hom-Hopf algebras. Then we study the relation between the Drinfel'd double $D(A)$ and Heisenberg double $H(A)=A\# A^{*}$, generalizing the main result in \cite{Lu}. Especially, the examples given in the paper are not obtained from the usual …
We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras.
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 $$ (R # H)# H^{d} \cong R \otimes (H # H^{d}) \hbox {as algebras in braided tensor category} {\cal C}.$$ In particular, we present two duality theorems for infinite braided Hopf algebras in the Yetter-Drinfeld module category.
Let $q$ be an $n^{th}$ root of unity for $n > 2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-Jürgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an $n$-dimensional associative algebra $A$. They further showed that each such module structure extends uniquely to make $A$ a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras $H$ "close" to the Taft algebras on finite-dimen…
For a quasi-triangular Hopf algebra $\left( H,R\right) $, there is a notion of transmuted braided group $H_{R}$ of $H$ introduced by Majid. The transmuted braided group $H_{R}$ is a Hopf algebra in the braided category $_{H}\mathcal{M}$. The $R$-adjoint-stable algebra associated with any simple left $H_{R}$-comodule is defined by the authors, and is used to characterize the structure of all irreducible Yetter-Drinfeld modules in ${}_{H}^{H} \mathcal{YD}$. In this note, we prove for a semisimple factorizable Hopf algebra $ \left( H,R\right) $ that any simple subcoalgebra of $H_R$ is $H$-stable …
The aim of this paper is to construct a new braided $T$-category via the generalized Yetter-Drinfel'd modules and Drinfel'd codouble over Hopf algebra, an approach different from that proposed by Panaite and Staic \cite{PS}. Moreover, in the case of finite dimensional, we will show that this category coincides with the corepresentation of a certain coquasitriangular Turaev group algebra that we construct. Finally we apply our theory to the case of group algebra.
To a finite Hopf-Galois extension $A | B$ we associate dual bialgebroids $S := \End_BA_B$ and $T := (A ø_B A)^B$ over the centralizer $R$ using the depth two theory in math.RA/0108067. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra \cite{KT,Doi} to depth two extensions using coring theory math.RA/0002105. Next we show that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem 5.1 in math.QA/9505024 for smash products with special modules over the Drinfel'd double, the Miy…
The relationship between comodules of a coring and flat connections is reviewed. In particular we specialise to corings which are built on a tensor product of algebra and a coalgebra. Such corings are in one-to-one correspondence with entwining structures, and their comodules are entwined modules. These include Yetter-Drinfeld and anti-Yetter-Drinfeld modules and their generalisations, hence all the modules of interest to Hopf-cyclic cohomology. In this way the interpretation of the latter as modules with flat connections [A Kaygun and M Khalkhali, Hopf modules and noncommutative differential …
The Hopf algebra $\mathcal{D}$ which is the subject of this paper can be viewed as a Drinfeld double of the bosonisation of the Jordan plane. Its prime and primitive spectra are completely determined. As a corollary of this analysis it is shown that $\mathcal{D}$ satisfies the Dixmier-Moeglin Equivalence, leading to the formulation of a conjecture on the validity of this equivalence for pointed Noetherian Hopf algebras.
We propose the notion of Hopf module algebras and show that the projection onto the subspace of coinvariants is an idempotent Rota-Baxter operator of weight -1. We also provide a construction of Hopf module algebras by using Yetter-Drinfeld module algebras. As an application, we prove that the positive part of a quantum group admits idempotent Rota-Baxter algebra structures.
We introduce the notion of Drinfeld twists for both set-theoretical YBE solutions and skew braces. We give examples of such twists and show that all twists between skew braces come from families of isomorphisms between their additive groups. We then describe the relation between these definitions and co-twists on FRT-type Hopf algebras in the category $\mathrm{SupLat}$, and prove that any co-twist on a co-quasitriangular Hopf algebra in $\mathrm{SupLat}$ induces a Drinfeld twist on its remnant skew brace. We go on to classify co-twists on bicrossproduct Hopf algebras coming from groups with un…
The faithful quasi-dual $H^d$ and strict quasi-dual $H^{d'}$ of an infinite braided Hopf algebra $H$ are introduced and it is proved that every strict quasi-dual $H^{d'}$ is an $H$-Hopf module. The connection between the integrals and the maximal rational $H^{d}$-submodule $H^{d rat}$ of $H^{d}$ is found. That is, $H^{d rat}\cong \int ^l_{H^d} \otimes H$ is proved. The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld category $(^B_B{\cal YD},C)$ are given.
We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n. We show that these quantum groups can be realized as Drinfel'd doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix and a quantum Casimir element. We discuss isomorphisms among these quantum groups and connections with multiparameter quantum groups.
This paper introduces group-cograded monoidal Hom-Hopf algebras, and shows that this kind of group-cograded monoidal Hom-Hopf algebras are monoidal Hom-Hopf algebras in the Turaev category $\mathcal{J}_{k}$ introduced by Canepeel and De Lombaerde. Then we define the $p$-Yetter-Drinfeld category over a group-cograded monoidal Hom-Hopf algebra, and construct a new kind of braided $T$-categories.
Let $H$ be a finite Hopf algebra with $C_{H,H} = C_{H,H}^{-1}.$ The duality theorem is shown for $H$, i.e., $$ (R # H)# H^{\hat *} \cong R \otimes (H \bar \otimes H^{\hat *}) \hbox {as algebras in} {\cal C}.$$ Also, it is proved that the Drinfeld double $(D(H),[b])$ is a quasi-triangular Hopf algebra in ${\cal C}$.
