For a group $\pi $, Turaev [12] introduced the notion of a braided $\pi $-monoidal category, here called Turaev braided $\pi$-category, and showed that such a category gives rise to a $3$-dimensional homotopy quantum field theory. Kirillov [5] found that such Turaev braided $\pi $-categories also provide a suitable mathematical tool to describe the orbifold models which arise in the study of conformal field theories. Virelizier [15] used Turaev braided $\pi$-category to construct Hennings-type invariants of flat $\pi $-bundles over complements of links in the $3$-sphere. We note that a Turaev braided $\pi$-category is a braided monoidal category when $\pi $ is trivial.

Starting from the category of Yetter-Drinfeld modules, Panaite and Staic [6] constructed a Turaev braided category over certain group $\pi$, generalizing the work of [7]. Turaev braided $\pi$-categories were further investigated by Panaite and Staic in [6], by Zunino [17]. In the present paper, let $G$ be the semi-direct product of the opposite group $\pi ^{op}$ of a group $\pi $ by $\pi$ and $A$ an $H$-bicomodule algebra. We first introduce a class of new categories ${} _{A}\mathcal {YD}^{H}_{G}$ as a disjoint union of family of categories $\{{}_{A}\mathcal {YD}^{H}(\alpha, \beta )\}_{(\alpha, \beta )\in G}$. Then we mainly show that the category $\{{}_{A}\mathcal {YD}^{H}(\alpha, \beta)\}_{(\alpha, \beta )\in G}$ forms a braided $T$-category, generalizing the main constructions by Panaite and Staic [6]. Finally, when $H$ is finite-dimensional we construct a quasitriangular $T$-coalgebra $\{ A\# H^*(\alpha, \beta )\}_{(\alpha, \beta )\in G}$, such that $\{{}_{A}\mathcal {YD}^{H}(\alpha, \beta)\}_{(\alpha, \beta )\in G}$ is isomorphic to the representation category of the quasitriangular $T$-coalgebra $\{ A\# H^*(\alpha, \beta )\}_{(\alpha, \beta )\in G}$.

The paper is organized as follows. In Section 3, let $G$ be the semi-direct product of the opposite group $\pi ^{op}$ of a group $\pi $ by $\pi$ and $A$ an $H$-bicomodule algebra, we first introduce a class of new categories ${} _{A}\mathcal {YD}^{H}_{G}$ as a disjoint union of family of categories $\{{}_{A}\mathcal {YD}^{H}(\alpha, \beta )\}_{(\alpha, \beta )\in G}$ and give necessary and sufficient conditions making ${} _{A}\mathcal {YD}^{H}_{G}$ into a braided $T$-category.

In Section 4, when $H$ is finite-dimensional, as an appliction, we construct a quasitriangular $T$-coalgebra $\{ A\# H^*(\alpha, \beta )\}_{(\alpha, \beta )\in G}$, such that $\{{}_{A}\mathcal {YD}^{H}(\alpha, \beta)\}_{(\alpha, \beta )\in G}$ is isomorphic to the representation category of the quasitriangular $T$-coalgebra $\{ A\# H^*(\alpha, \beta )\}_{(\alpha, \beta )\in G}$.

Throughout the paper, we let $\mathbb{k}$ be a fixed field and denote by $\otimes $ the the tensor product over $\mathbb{k}$. For the comultiplication $\Delta $ in a coalgebra $C$, we use the Sweedler-Heyneman's notation [12]:

$ \Delta(c)=c_{1}\otimes c_{2} $

for any $c\in C$. For a left $C$-comodule $(M, \rho ^l)$ and a right $C$-comodule $(N, \rho ^r)$, we write

$ \rho^l(m)=m_{(-1)}\otimes m_{(0)} \quad \mbox{and}\quad \rho^r(n)=n_{(0)}\otimes n_{(1)}, $

respectively, for all $m\in M$ and $n\in N$. For a Hopf algebra $A$, we always denote by $Aut(A)$ the group of Hopf automorphisms of $A$.

Let $\pi $ be a group with the unit $e$. We recall that a Turaev $\pi $-category (see [12]) is a monoidal category ${\cal C}$ which consists of the following data.

A family of subcategories $\{{\cal C}_{\alpha }\}_{\alpha \in \pi }$ such that ${\cal C}$ is a disjoint union of this family and such that $U\otimes V\in {\cal C}_{\alpha \beta }$, for any $\alpha, \beta \in \pi $, if the $U\in {\cal C}_{\alpha }$ and $V\in {\cal C}_{\beta }$. Here the subcategory ${\cal C}_{\alpha }$ is called the $\alpha $th component of ${\cal C}$.

A group homomorphism $\varphi : \pi \to {\rm{aut}}({\cal C}), \beta \mapsto \varphi _{\beta }$, the conjugation, (where aut $({\cal C})$ is the group of invertible strict tensor functors from ${\cal C}$ to itself) such that $\varphi _{\beta }({\cal C}_{\alpha })={\cal C}_{\beta \alpha \beta ^{-1}}$ for any $\alpha, \beta \in \pi $. Here the functors $\varphi _{\beta }$ are called conjugation isomorphisms.

We will use the left index notation in [12] or [17]: given $\beta \in G$ and an object $V\in {\cal C}_{\beta }$, the functor $\varphi _{\beta }$ will be denoted by ${}^V( \cdot )$ or ${}^{\beta }( \cdot )$. We use the notation ${}^\bar V( \cdot )$ for ${}^{\beta ^{-1}} ( \cdot )$. Then we have ${}^V id_U=id_{^{ V } U}$ and ${}^V(g\circ f)={}^Vg\circ {}^Vf$. We remark that since the conjugation $\varphi : G \to {\rm{aut}}({\cal C})$ is a group homomorphism, for any $V, W\in {\cal C}$, we have ${}^{V\otimes W}( \cdot ) ={}^V({}^W( \cdot ))$ and ${}^1( \cdot )={}^V({}^\bar V( \cdot )) ={}^\bar V({}^V( \cdot ))=id_{\cal C}$ and that since, for any $V\in {\cal C}$, the functor ${}^V( \cdot )$ is strict, we have ${}^V(f\otimes g)={}^Vf\otimes {}^Vg$, for any morphism $f$ and $g$ in ${\cal C}$, and ${}^V1=1$. And we will use ${\cal C}(U, V)$ for the set of morphisms (or arrows) from $U$ to $V$ in ${\cal C }$.

Recall from [12] that a braided crossed category is a crossed category ${\cal C}$ endowed with a braiding, i.e., with a family of isomorphisms

$ \tau =\{\tau _{U, V}\in {\cal C}(U\otimes V, ({}^UV)\otimes U)\}_{U, V\in {\cal C}} $

satisfying the following conditions:

$ \begin{eqnarray} &&\mbox {for any arrow} \, f\in {\cal C}_{\alpha }(U, U') \, \, \mbox {with}\, \alpha \in \pi, g\in {\cal C}(V, V'), \, \mbox {we have} \nonumber\\ && \, \, (({}^{\alpha }g)\otimes f)\circ \tau _{U, V}=\tau _{U' V'}\circ (f\otimes g);\end{eqnarray} $

(2.1)

$\begin{eqnarray} & &\mbox {for all} \, \, U, V, W\in {\cal C}, \, \mbox {we have} \nonumber\\ &&\quad \tau _{U\otimes V, W}=a_{{}^{U\otimes V}W, U, V}\circ (\tau _{U, {}^VW\otimes V})\circ a^{-1}_{U, {}^VW, V}\circ (\mathit{\iota } _U\otimes \tau _{V, W}) \circ a_{U, V, W}, \quad \quad \quad \quad \end{eqnarray} $

(2.2)

$\begin{eqnarray} &&\quad \tau _{U, V\otimes W}=a^{-1}_{{}^UV, {}^UW, U} \circ (\mathit{\iota } _{({}^UV)}\otimes \tau _{U, W})\circ a_{{}^UV, U, W}\circ (\tau _{U, V}\otimes \mathit{\iota }_W)\circ a^{-1}_{U, V, W}, \end{eqnarray} $

(2.3)

$\begin{eqnarray} & &\mbox {for any}\, \, U, V\in {\cal C}, \alpha \in \pi, \varphi _{\alpha }(\tau _{U, V})=\tau _{\varphi _{\alpha }(U), \varphi _{\alpha }(V)}. \end{eqnarray} $

(2.4)

In this paper, we use terminology as in Zunino [17]; for the subject of Turaev categories, see also the original paper of Turaev [12]. If ${\cal C}$ is a braided crossed category then we call ${\cal C}$ a braided $T$-category.

Let $\pi $ be a group with unit $e$. Recall from Turaev [12] that a $\pi $-coalgebra is a family of $k$-spaces $C=\{C_{\alpha }\}_{\alpha \in \pi }$ together with a family of $k$-linear maps $\Delta =\{\Delta _{\alpha, \beta}: C_{\alpha \beta}\longrightarrow C_{\alpha }\otimes C_{\beta }\}_{\alpha, \beta\in \pi }$ (called the comultiplication) and a $k$-linear map $\varepsilon : C_e\longrightarrow \mathbb{k}$ (called the counit), such that $\Delta $ is coassociative in the sense that,

$ \begin{eqnarray} && (\Delta _{\alpha, \beta}\otimes id_{C_{\lambda }})\Delta _{\alpha \beta, \lambda}= (id_{C_{\alpha }}\otimes \Delta _{\beta, \lambda })\Delta _{\alpha, \beta \lambda} \, \, \mbox {for any}\, \, \alpha, \beta, \lambda \in \pi, \quad \quad \quad \quad \quad \quad \quad\end{eqnarray} $

(2.5)

$\begin{eqnarray} && (id_{C_{\alpha }}\otimes \varepsilon )\Delta _{\alpha, e}=id_{C_{\alpha }}=(\varepsilon \otimes id_{C_{\alpha }})\Delta _{e, \alpha } \mbox { for all }\alpha \in \pi. \end{eqnarray} $

(2.6)

We use the Sweedler-like notation (see [14]) for a comultiplication in the following way: for any $\alpha, \beta \in \pi $ and $c\in C_{\alpha \beta }$, we write $ \Delta _{\alpha, \beta }(c)=c_{(1, \alpha )}\otimes c_{(2, \beta )}. $

A $T-coalgebra$ is a $\pi $-coalgebra $H=(\{H_{\alpha }\}, \Delta, \varepsilon )$ together with a family of $k$-linear maps $S=\{S_{\alpha }: H_{\alpha }\longrightarrow H_{\alpha ^{-1}}\}_{\alpha \in \pi }$ (called the antipode), and a family of algebra isomorphisms $\varphi =\{\varphi _{\beta }: H_{\alpha } \to H_{\beta \alpha \beta ^{-1}}\}_{\alpha, \beta \in \pi}$ (called the crossing) such that

$ {\rm{each}}\;{H_\alpha }\;{\rm{is}}\;{\rm{an}}\;{\rm{algebra}}\;{\rm{with}}\;{\rm{multiplication}}\;{m_\alpha }\;{\rm{and}}\;{\rm{unit}}\;{1_\alpha } \in {H_\alpha }, {\rm{ }} $

(2.7)

${\rm{for}}\;{\rm{all}}\;\alpha, \beta \in \pi, \quad {\Delta _{\alpha, \beta }}\;{\rm{and}}\;\varepsilon :{H_e} \to k\;{\rm{are}}\;{\rm{algebra}}\;{\rm{maps}}, $

(2.8)

${\rm{for}}\;\mathit{\alpha } \in \pi, {m_\alpha }({S_{{\alpha ^{ - 1}}}} \otimes i{d_{{H_\alpha }}}){\Delta _{{\alpha ^{ - 1}}, \alpha }} = \varepsilon {1_\alpha } = {m_\alpha }(i{d_{{H_\alpha }}} \otimes {S_{{\alpha ^{ - 1}}}}){\Delta _{\alpha, {\alpha ^{ - 1}}}}, \quad $

(2.9)

${\rm{for}}\;{\rm{all}}\;\alpha, \beta, \gamma \in \pi, \quad ({\varphi _\beta } \otimes {\varphi _\beta }){\Delta _{\alpha, \gamma }} = {\Delta _{\beta \alpha {\beta ^{ - 1}}, \beta \alpha {\beta ^{ - 1}}}}{\varphi _\beta }, \quad $

(2.10)

${\rm{for}}\;{\rm{all}}\;\beta \in \pi, \quad \varepsilon {\varphi _\beta } = \varepsilon \quad $

(2.11)

${\rm{for}}\;{\rm{all}}\;\alpha, \beta \in \pi, \quad {\varphi _\alpha }{\varphi _\beta } = {\varphi _{\alpha \beta }}. $

(2.12)

Let $G, L$ be two groups and $G$ act on the left the group $L$ by automorphisms. Then $L\times G$ is a group with the multiplication

$ (l, g)(l', g')=(l (g\rhd l'), gg'), $

which is called a semi-direct product of $L$ by $G$ and denoted by $L\ltimes G$. A group $\pi $ is a semi-direct product of $L$ by $G$ if and only if $L$ is a normal subgroup of $\pi $, $G$ is a subgroup of $\pi $, $L\cap G=1$, and $\pi =LG$ (see [16]).

Let $\pi $ be a group and let $L=\pi ^{op}$, the opposite group of a group $\pi $. Consider the adjoint action of $\pi $ on $L$ by defining: $\gamma \rhd \alpha =\gamma \alpha \gamma ^{-1}$ for all $\alpha, \gamma \in \pi $. Then we have the semi-direct product $\pi ^{op}\ltimes \pi $. The opposite group $(\pi ^{op}\ltimes \pi )^{op}$ of the group $\pi ^{op}\ltimes \pi $ is denoted by $G$ with the multiplication, for all $\alpha, \beta, \lambda, \gamma \in \pi $:

$ \begin{equation} (\alpha, \beta )\#(\lambda, \gamma )= (\gamma \alpha \gamma ^{-1}\lambda, \gamma \beta ), \end{equation} $

(2.13)

which was called a twisted semi-direct square of group $\pi $ (see [16]). Moreover $\pi $ is a subgroup of $G$ and $(\alpha, \beta )^{-1}=(\beta ^{-1}\alpha ^{-1}\beta, \beta ^{-1})$.

Definition 3.1  Let $H$ be a Hopf algebra with bijective antipode $S$ and $\pi $ a group with the unit $e$. Let $A$ be an $H$-bicomodule algebra and $\zeta_{\alpha}, \zeta_\beta\in Aut(H)$. An $(\alpha, \beta )$-quantum Yetter-Drinfeld module is a vector space $M$, such that $M$ is a left $A$-module (with notation $a\otimes m\mapsto a\cdot m$) and right $H$-comodule (with notation $M\rightarrow M\otimes H, m\mapsto m_{(0)}\otimes m_{(1)} $) with the following compatibility condition:

$ \begin{equation} \rho(a\cdot m)=a_{(0)}\cdot m_{(0)} \otimes \zeta_\beta (a_{(1)})m_{(1)} \zeta_\alpha (S^{-1}(a_{(-1)})) \end{equation} $

(3.1)

for all $a\in A$ and $m\in M$. We denote by ${}_A{\cal {YD}}^{H}(\alpha, \beta )$ the category of $(\alpha, \beta )$-quantum Yetter-Drinfeld module, morphisms being $A$-linear and $H$-colinear maps.

Remark 3.2  Let $H$ be a Hopf algebra with bijective antipode $S$ and $\zeta_{\alpha}, \zeta_\beta\in Aut(H).$ Let $A$ be an $H$-bicomodule algebra. Then $A\otimes H$ is an object in ${}_A{\cal {YD}}^{H}(\alpha, \beta )$ with the following structures:

$ \begin{eqnarray*} a\cdot (b \otimes h)=ab\otimes h, \quad \quad \rho (b\otimes h )= b_{(0)}\otimes h_1\otimes \zeta_\beta (b_{(1)})h_2 \zeta_\alpha (S^{-1}(b_{(-1)})) \end{eqnarray*} $

for all $a, b\in A$ and $h\in H$. Furthermore, if $A$ and $H$ are bialgebras, then it is easy to check $\Bbbk$ is an object in ${}_A{\cal {YD}}^{H}(\alpha, \beta )$ with structures: $ a\cdot x=\varepsilon _A(a)x$ and $\rho (x)=x\otimes 1_H$, for all $x\in \Bbbk$ if and only if the following condition holds:

$ \begin{equation} \varepsilon _A(a)1_H=\varepsilon _A(a_{(0)})\zeta_\alpha(a_{(-1)})\zeta_\beta(S^{-1}(a_{(1)})) \end{equation} $

(3.2)

for all $a\in A$.

Thus we have a Turaev $G$-category ${}_{A}{\cal YD}^H_{G}$ as a disjoint union of family of categories $\{{}_{H}{\cal YD}^H_{G}(\alpha, \beta )\}_{(\alpha, \beta )\in G}$ over the family of the left-right $(\alpha, \beta)$-Yetter-Drinfeld modules, with $(\alpha, \beta )\in G$.

Example 3.3  (1) Let $H$ be a Hopf algebra with a bijective antipode and $\zeta : \pi \to Aut(H)$ a group homomorphism. Then category $ _{H}\mathcal {YD}^{H}(\alpha, \beta)$ is actually the category of $(\alpha, \beta )$-Yetter-Drinfel'd modules studied in Panaite and Staic [6].

(2) For $\zeta_\alpha=\zeta_\beta=id_{H}$, we have $ _{H}\mathcal {YD}^{H}(id, id)={}_H{\cal YD}^H$, the usual quantum Yetter-Drinfel'd module category in the sense of Caenepeel et al. [2].

(3) For $\zeta_\alpha=S^2, \zeta_\beta=id_{H}$, the compatibility condition (2.1) becomes

$ (a\cdot m)_{(0)}\otimes (a\cdot m)_{(1)}=a_{(0)}\cdot m_{(0)}\otimes a_{(-1)}m_{(1)}S(a_{(1)}), $

hence $_{H}\mathcal {YD}^{H}(S^2, id)$ is the usual anti-quantum Yetter-Drinfeld module category in the sense of Caenepeel et al. [2].

The following notion is a generalization of one in Panaite and Staic in [6].

Proposition 3.4  For any $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta )$ and $N\in {}_A{\cal {YD}}^{H}(\gamma, \delta )$, then we have $M \otimes N \in {}_A{\cal {YD}}^{H}(\delta \alpha \delta ^{-1}\gamma, \delta \beta)$ with structures:

$ \begin{eqnarray*} && a\triangleright (m \otimes n)=a_2\cdot m \otimes a_1\cdot n, \\ && \rho _{M\otimes N}(m\otimes n ):=(m\otimes n)_{\langle0\rangle}\otimes (m\otimes n)_{\langle1\rangle}=m_{(0)}\otimes n_{(0)}\otimes \zeta_\delta (m_{(1)})\zeta_{\delta \alpha \delta^{-1}}(n_{(1)}) \end{eqnarray*} $

if and only if the following condition holds:

$ \begin{eqnarray} &&a_{2(0)}\otimes a_{1(0)}\otimes \zeta_{\delta \beta}(a_{2(1)})\zeta_{\delta}(c)\zeta_{\delta\alpha}(S^{-1}(a_{2(-1)})) \zeta_{\delta \alpha}(a_{1(1)}) \zeta_{\delta \alpha\delta^{-1}}(d) \zeta_{\delta \alpha\delta^{-1}\gamma}(S^{-1}(a_{1(-1)})))\nonumber\\ &=&a_{(0)2}\otimes a_{(0)1} \otimes \zeta_{\delta\beta}(a_{(1)})\zeta_{\delta}(c)\zeta_{\delta\alpha\delta^{-1}}(d)\zeta_{\delta\alpha\delta^{-1}\gamma}(S^{-1}(a_{-1})). \end{eqnarray} $

(3.3)

Proof  It is easy to see that $M\otimes N$ is a left $A$-module and $M\otimes N$ is a right $H$-comodule. We compute the compatibility condition:

$ \begin{eqnarray*} &&\rho _{M\otimes N}(a\triangleright (m\otimes n))\\ &=&(a_2\cdot m)_{(0)} \otimes (a_1\cdot n)_{(0)}\otimes \zeta_\delta ((a_{2}\cdot m)_{(1)}) \zeta_{\delta \alpha \delta^{-1}}((a_1\cdot n)_{(1)})\\ &=&a_{2(0)}\cdot m_{(0)}\otimes a_{1(0)}\cdot n_{(0)}\otimes \zeta_{\delta} (\zeta_{\beta}(a_{2(1)})m_{(1)}\zeta_{\alpha}(S^{-1}(a_{2(-1)})) )\\ &&\zeta_{\delta \alpha\delta^{-1}}(\zeta_{\delta}(a_{1(1)})n_{(1)}\zeta_{\gamma}(S^{-1}(a_{1(-1)})))\\ &=&a_{2(0)}\cdot m_{(0)}\otimes a_{1(0)}\cdot n_{(0)}\otimes \zeta_{\delta \beta}(a_{2(1)})\zeta_{\delta}(m_{(1)})\zeta_{\delta\alpha}(S^{-1}(a_{2(-1)})) \zeta_{\delta \alpha}(a_{1(1)})\\ &&\zeta_{\delta \alpha\delta^{-1}}(n_{(1)}) \zeta_{\delta \alpha\delta^{-1}\gamma}(S^{-1}(a_{1(-1)})))\\ &\stackrel{(3.3)}{=}& a_{(0)2}\cdot m_{(0)}\otimes a_{(0)1}\cdot n_{(0)} \otimes \zeta_{\delta\beta}(a_{(1)})\zeta_{\delta}(m_{(1)})\zeta_{\delta\alpha\delta^{-1}}(n_{(1)})\zeta_{\delta\alpha\delta^{-1}\gamma}(S^{-1}(a_{-1}))\\ &=&a_{(0)}\cdot (m\otimes n)_{\langle0\rangle} \otimes \zeta_{\delta\beta}(a_{(1)})(m\otimes n)_{\langle1\rangle}\zeta_{\delta\alpha\delta^{-1}\gamma}(S^{-1}(a_{-1})) \end{eqnarray*} $

and this shows that $M \otimes N \in {}_A{\cal {YD}}^{H}(\delta \alpha \delta ^{-1}\gamma, \delta \beta)$.

Conversely, by Remark 3.2, since $A\otimes H\in {}_A{\cal {YD}}^{H}(\alpha, \beta )$, we let $m=1\otimes c$ and $n=1\otimes d$ for any $c, d\in H$ and easily get eq. (3.3).

The following proposition is straightforward.

Proposition 3.5  For any $N \in {}_A{\cal {YD}}^{H}(\gamma, \delta )$ and $(\alpha, \beta )\in G$. Define ${}^{(\alpha, \beta)}N=N$ as vector space, with structures

$ \begin{eqnarray*} && a\blacklozenge n=\xi _{\alpha^{-1}\beta}(a)\cdot n, \\ &&\rho (n):=n_{[0]}\otimes n_{[1]}= n_{(0)}\otimes \zeta _{\beta^{-1}\delta \alpha \delta^{-1}}(n_{(1)}). \end{eqnarray*} $

Then we have that ${}^{(\alpha, \beta)}N \in {}_A{\cal {YD}}^{H}((\alpha, \, \beta)\# (\gamma, \, \delta)\# (\alpha, \, \beta)^{-1})$ if and only if the following condition holds:

$ \begin{eqnarray} && \xi _{\alpha^{-1}\beta}(a)_{(0)}\otimes \zeta _{\beta^{-1}\delta\alpha\delta^{-1}}( \zeta _{\delta}(\xi _{\alpha^{-1}\beta}(a)_{(-1)}) c\zeta_{\gamma} (S^{-1}(\xi _{\alpha^{-1}\beta}(a)_{(-1)})))\nonumber\\ & =& \xi _{\alpha^{-1}\beta }(a_{(0)})\otimes \zeta_{\beta^{-1}\delta\beta}(a_{(-1)}) \xi _{\beta^{-1}\delta\alpha\delta^{-1}}(c)\zeta_{\beta^{-1}\delta\alpha\delta^{-1}\gamma\delta^{-1}\beta}(S^{-1}(a_{(1)})). \end{eqnarray} $

(3.4)

Furthermore, let $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$ and $(\mu, \nu)\in G$. Then we have

$ {}^{(\alpha, \, \beta)\#(\mu, \, \nu)}N={}^{(\alpha, \, \beta)}({}^{(\mu, \, \nu)}N), \quad \quad {}^{(\mu, \, \nu)}(M\otimes N)= {}^{(\mu, \, \nu)}M \otimes {}^{(\mu, \, \nu)}N. $

Proof  We only show the first claim as follows.

By Remark 3.2, $A\otimes H\in {}_A{\cal {YD}}^{H}(\gamma, \delta )$ for any $(\gamma, \delta )\in G$. For any $d\in H$, then we have $ (a\blacklozenge (1\otimes d))_{[0]}\otimes (a\blacklozenge (1\otimes d))_{[1]} =a_{[0]}\blacklozenge (1\otimes d)_{[0]}$ $ \otimes \zeta_{\beta^{-1}\delta\beta}(a_{(-1)})(1\otimes d)_{[1]}\zeta_{\beta^{-1}\delta\alpha\delta^{-1}\gamma\delta^{-1}\beta}(S^{-1}(a_{(1)})), $ which implies eq. (3.4).

Conversely, one has

$ \begin{eqnarray*} &&(a\blacklozenge n)_{[0]}\otimes (a\blacklozenge n)_{[1]}\\ &=&(\xi _{\alpha^{-1}\beta}(a)\cdot n)_{(0)}\otimes \zeta _{\beta^{-1}\delta \alpha \delta^{-1}}((\xi _{\alpha^{-1}\beta}(a)\cdot n)_{(1)})\\ &=&\xi _{\alpha^{-1}\beta}(a)_{(0)}\cdot n_{(0)}\otimes \zeta _{\beta^{-1}\delta\alpha\delta^{-1}}( \zeta _{\delta}(\xi _{\alpha^{-1}\beta}(a)_{(-1)}) n_{(1)}\zeta_{\gamma} (S^{-1}(\xi _{\alpha^{-1}\beta}(a)_{(-1)})))\\ &\stackrel{(3.4)}{=}& \xi _{\alpha^{-1}\beta }(a_{(0)})\cdot n_{(0)}\otimes \zeta_{\beta^{-1}\delta\beta}(a_{(-1)}) \xi _{\beta^{-1}\delta\alpha\delta^{-1}}(n_{(1)})\zeta_{\beta^{-1}\delta\alpha\delta^{-1}\gamma\delta^{-1}\beta}(S^{-1}(a_{(1)}))\\ &=&a_{(0)}\blacklozenge n_{[0]}\otimes \zeta_{\beta^{-1}\delta\beta}(a_{(-1)}) n_{[1]}\zeta_{\beta^{-1}\delta\alpha\delta^{-1}\gamma\delta^{-1}\beta}(S^{-1}(a_{(1)})) . \end{eqnarray*} $

Now define a group homomorphism $\varphi : G \to Aut({}_A{\cal {YD}}^{H}_{G})$, $(\alpha, \beta )\mapsto \varphi _{(\alpha, \beta )}$, as

$ \varphi _{(\alpha, \beta)}: {}_A{\cal {YD}}^{H}(\gamma, \delta ) \to {}_A{\cal {YD}}^{H} ((\alpha, \beta )\#(\gamma, \delta )\#(\alpha, \beta )^{-1}), \varphi _{\alpha, \beta }(N)={}^{(\alpha, \beta )}N, $

and the functor $\varphi _{(\alpha, \beta)}$ acts as identity on morphisms.

Consider now a map $\mathscr{Q}: \, \, H\otimes H\rightarrow A\otimes A$ with a twisted convolution inverse $\mathscr{R}$, that means that

$ \mathscr{Q}(h_2\otimes g_2)\mathscr{R}(h_1\otimes g_1)=\varepsilon (h)1_A\otimes \varepsilon(g)1_A $

for all $h, g\in H$. Sometimes, we write $\mathscr{Q}(h\otimes g):=\mathscr{Q}^1(h\otimes g)\otimes \mathscr{Q}^2(h\otimes g)$ for all $h, g\in H$.

For any $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$, $N\in {}_A{\cal {YD}}^{H}(\gamma, \delta)$ and $P \in {}_A{\cal {YD}}^{H}(\mu, \nu)$. Define a map as follows:

$ \begin{eqnarray} &&c_{M, N}: M \otimes N \rightarrow {}^{M}N \otimes M, \nonumber\\ &&c_{M, N}(m\otimes n)=\mathscr {Q}(n_{(1)}\otimes \zeta _{\alpha^{-1}}(m_{(1)}))(n_{(0)}\otimes m_{(0)}). \end{eqnarray} $

(3.5)

In what follows, our main aim is to give some necessary and sufficient conditions on $\mathscr{Q}$ such that the $c_{M, N}$ defines a braiding on ${}_A{\cal {YD}}^{H}_{G}$. For this, we will find conditions under which $c_{M, N}$ is both $A$-linear and $H$-colinear, and the following conditions hold:

$ \begin{eqnarray} && c_{M\otimes N, P}=(c_{M, \, \, {{}^{N}P}}\otimes id_{N})\circ (id_{M}\otimes c_{N, P}), \end{eqnarray} $

(3.6)

$ \begin{eqnarray} && c_{M, N\otimes P}=(id_{ {}^{M} N} \otimes c_{M, P} )\circ (c_{M, N}\otimes id_{P}). \end{eqnarray} $

(3.7)

Furthermore, if $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$ and $N \in {}_A{\cal {YD}}^{H}(\gamma, \delta)$, then we want to show the following:

$ \begin{equation} c_{{}^{(\mu, \nu)}M, \, {}^{(\mu, \nu)}N}=c_{M, N} \end{equation} $

(3.8)

holds, for any $(\mu, \nu)\in G$.

In order to approach to our main result we need some lemmas.

Lemma 3.6  For any $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$ and $N \in {}_A{\cal {YD}}^{H}(\gamma, \delta)$. Then $c_{M, N}$ is $A$-linear if and only if the following condition is satisfied:

$ \begin{eqnarray} &&\mathscr{Q}(\zeta_{\delta}(a_{1(-1)})d\zeta_{\gamma}(S^{-1}(a_{1(1)}))\otimes \zeta _{\alpha^{-1}}(\zeta_{\beta}(a_{2(-1)})c\zeta_{\alpha}(S^{-1}(a_{2(1)}))) (a_{1(0)}\otimes a_{2(0)})\nonumber\\ &=&[(\xi _{\alpha^{-1}\beta} \otimes 1)\Delta ^{cop}(a)]\mathscr{Q} (d\otimes \zeta _{\alpha^{-1}}(c)) \end{eqnarray} $

(3.9)

for all $a\in A$ and $c, d\in H$.

Proof  If $c_{M, N}$ is $A$-linear then it is easy to get

$ a\cdot c_{M, N}(m\otimes n) =[(\xi _{\alpha^{-1}\beta} \otimes 1)\Delta ^{cop}(a)]\mathscr{Q}(n_{(1)} \otimes \zeta _{\alpha^{-1}}(m_{(1)}))(n_{(0)}\otimes m_{(0)}) $

and

$ \begin{eqnarray*} &&c_{M, N}(a\cdot (m\otimes n))\\ &=&\mathscr{Q}(\zeta_{\delta}(a_{1(-1)})n_{(1)}\zeta_{\gamma}(S^{-1}(a_{1(1)}))\otimes \zeta _{\alpha^{-1}}(\zeta_{\beta}(a_{2(-1)})m_{(1)}\zeta_{\alpha}(S^{-1}(a_{2(1)})))\\ && (a_{1(0)}\cdot n_{(0)}\otimes a_{2(0)}\cdot m_{(0)}). \end{eqnarray*} $

Considering these equations and taking $M=N=A\otimes C$ and $m=1\otimes c$ and $n=1\otimes d$ for all $c, d\in H$. Then we can get eq. (3.9).

Conversely, by the above formulas it is easy to see that $c_{M, N}$ is $A$-linear.

Lemma 3.7  For any $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$ and $N \in {}_A{\cal {YD}}^{H}(\gamma, \delta)$. Then $c_{M, N}$ is $H$-colinear if and only if the following condition is satisfied:

$ \begin{eqnarray} &&\mathscr{Q}(d_1\otimes \zeta _{\alpha^{-1}}(c_1)) \otimes \zeta _{\delta }(c_2)\zeta _{\delta \alpha \delta^{-1}}(d_2)=\mathscr{Q}^1(d_2\otimes \zeta _{\alpha^{-1}}(c_2))_{(0)}\otimes \mathscr{Q}^2(d_2\otimes \zeta _{\alpha^{-1}}(c_2))_{(0)} \nonumber\\ &&\otimes \zeta _{\delta \alpha \delta^{-1}}(\zeta_{\delta} (\mathscr{Q}^1(d_{2}\otimes \zeta _{\alpha^{-1}}(c_{2}))_{(-1)})d_{1}\zeta_{\gamma}(S^{-1}(\mathscr{Q}^1(d_{2}\otimes \zeta _{\alpha^{-1}}(c_{2}))_{(1)})))\nonumber\\ && \otimes \zeta _{\delta\alpha\delta^{-1}\gamma\alpha^{-1}}(\zeta_{\alpha} (\mathscr{Q}^2(d_{2}\otimes \zeta _{\alpha^{-1}}(c_{2}))_{(-1)})c_{1}\zeta_{\alpha}(S^{-1}(\mathscr{Q}^2(d_{2}\otimes \zeta _{\alpha^{-1}}(c_{2}))_{(1)}))) \end{eqnarray} $

(3.10)

for all $c, d\in H$.

Proof  If $c_{M, N}$ is $H$-colinear then we do the following calculations:

$ \begin{eqnarray*} &&\rho \circ c_{M, N}(m\otimes n)\\ &=&(\mathscr{Q}^1(n_{(1)}\otimes \zeta _{\alpha^{-1}}(m_{(1)}))\cdot n_{(0)})_{\langle0\rangle}\otimes (\mathscr{Q}^2(n_{(1)}\otimes \zeta _{\alpha^{-1}}(m_{(1)})) \cdot m_{(0)})_{(0)} \\ &&\otimes (\zeta _{\beta }(\mathscr{Q}^1(n_{(1)}\otimes \zeta _{\alpha^{-1}}(m_{(1)}))\cdot n_{(0)})_{\langle1\rangle}) (\zeta _{\delta\alpha\delta^{-1}\gamma\alpha^{-1}}(\mathscr{Q}^2(n_{(1)}\otimes \alpha^{-1}\cdot m_{(1)})\cdot m_{(0)})_{(1)})\\ &=&(\mathscr{Q}^1(n_{(1)}\otimes \zeta _{\alpha^{-1}}(m_{(1)}))\cdot n_{(0)})_{(0)}\otimes (\mathscr{Q}^2(n_{(1)}\otimes \zeta _{\alpha^{-1}} (m_{(1)}))\cdot m_{(0)})_{(0)}\\ &&\otimes (\zeta _{\delta \alpha \delta^{-1}}(\mathscr{Q}^1(n_{(1)}\otimes \zeta _{\alpha^{-1}}(m_{(1)}))\cdot n_{(0)})_{(1)}) (\zeta _{\delta\alpha\delta^{-1}\gamma\alpha^{-1}} (\mathscr{Q}^2(n_{(1)}\otimes \zeta _{\alpha^{-1}}(m_{(1)}))\cdot m_{(0)})_{(1)})\\ &=&(\mathscr{Q}^1(n_{(1)2}\otimes \zeta _{\alpha^{-1}}(m_{(1)2}))_{(0)}\cdot n_{(0)})\otimes (\mathscr{Q}^2(n_{(1)2}\otimes \zeta _{\alpha^{-1}} (m_{(1)2}))_{0}\cdot m_{(0)}) \\ &&\otimes \zeta _{\delta \alpha \delta^{-1}}(\zeta_{\delta} (\mathscr{Q}^1(n_{(1)2}\otimes \zeta _{\alpha^{-1}}(m_{(1)2}))_{(-1)})n_{(1)1}\zeta_{\gamma}(S^{-1}(\mathscr{Q}^1(n_{(1)2}\otimes \zeta _{\alpha^{-1}}(m_{(1)2}))_{(1)}))) \\ &&\otimes \zeta _{\delta\alpha\delta^{-1}\gamma\alpha^{-1}}(\zeta_{\alpha} (\mathscr{Q}^2(n_{(1)2}\otimes \zeta _{\alpha^{-1}}(m_{(1)2}))_{(-1)})m_{(1)1}\zeta_{\alpha}(S^{-1}(\mathscr{Q}^2(n_{(1)2}\otimes \zeta _{\alpha^{-1}}(m_{(1)2}))_{(1)}))) \end{eqnarray*} $

and

$ \begin{eqnarray*} &&(c_{M, N}\otimes id)\circ \rho (m\otimes n)=c_{M, N}(m\otimes n)_{\langle0\rangle}\otimes (m\otimes n)_{\langle1\rangle}\\ & =&\mathscr{Q}(n_{(1)1}\otimes \zeta _{\alpha^{-1}}(m_{(1)1}))(n_{(0)}\otimes m_{(0)}) \otimes \zeta _{\delta }(m_{(1)2})\zeta _{\delta \alpha \delta^{-1}}(n_{(1)2}). \end{eqnarray*} $

Now we let $M=N=A\otimes H$ and take $m=1\otimes c$ and $n=1\otimes d$ for all $c, d\in H$. Then we can get eq. (3.10).

Conversely, by the above formulas it is easy to see that $c_{M, N}$ is $H$-colinear.

Lemma 3.8  For any $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$, $N \in {}_A{\cal {YD}}^{H}(\gamma, \delta)$ and $P \in {}_A{\cal {YD}}^{H}(\mu, \nu)$. Then eq. (3.6) holds if and only if the following condition is satisfied, with $\mathscr{U}=\mathscr{Q}$:

$ \begin{eqnarray} &&(id\otimes \Delta^{cop})\mathscr{Q}(h\otimes \zeta _{\gamma^{-1}}(\zeta _{\delta \alpha^{-1}} (c)d))=[(\xi _{\gamma ^{-1} \delta } \otimes 1)\mathscr{U}( \zeta_{\delta^{-1}\nu \delta}(\mathscr{Q}^1(h_{2}\otimes \zeta _{\gamma^{-1}} (d))_{(-1)}) \nonumber\\ && h_{1} \zeta_{\delta^{-1}\nu \gamma \nu ^{-1}\mu \gamma ^{-1}\delta}(S^{-1}(\mathscr{Q}^1(h_{2}\otimes \zeta _{\gamma^{-1}} (d))_{(1)}))\otimes \zeta _{\alpha ^{-1}}(c))]\nonumber\\ && (\mathscr{Q}^1(h_{2}\otimes \zeta _{\gamma^{-1}} (d))_{(0)}\otimes 1)\otimes \mathscr{Q}^2(h_{2}\otimes \zeta _{\gamma^{-1}} (d)) \end{eqnarray} $

(3.11)

for all $c, d, h\in H$.

Proof  If eq. (3.6) holds. Then we compute as follows:

$ \begin{eqnarray*} &&(c_{M, \, \, {{}^{N}P}}\otimes id_{N})\circ (id_{M}\otimes c_{N, P})(m\otimes n\otimes p)\\ &=&\mathscr{U}((\mathscr{Q}^1(p_{(1)}\otimes \zeta _{\gamma^{-1}}(n_{(1)}))\cdot p_{(0)})_{(1)}\otimes \zeta _{\alpha ^{-1}}(m_{(1)})) ((\mathscr{Q}^1(p_{(1)}\otimes \zeta _{\gamma^{-1}}(n_{(1)}))\cdot p_{(0)})_{(0)}\\ && \otimes m_{(0)}\otimes \mathscr{Q}^2(p_{(1)}\otimes \zeta _{\gamma^{-1}}(n_{(1)}))\cdot n_{(0)})\\ &=&[(\xi _{\gamma ^{-1} \delta } \otimes 1)\mathscr{U}( \zeta_{\delta^{-1}\nu \delta}(\mathscr{Q}^1(p_{(1)2}\otimes \zeta _{\gamma^{-1}} (n_{(1)}))_{(-1)}) p_{(1)1} \zeta_{\delta^{-1}\nu \gamma \nu ^{-1}\mu \gamma ^{-1}\delta}\\ && (S^{-1}(\mathscr{Q}^1(p_{(1)2}\otimes \zeta _{\gamma^{-1}} (n_{(1)}))_{(1)}))\otimes \zeta _{\alpha ^{-1}}(m_{(1)}))] \\ && (\mathscr{Q}^1(p_{(1)2}\otimes \zeta _{\gamma^{-1}} (n_{(1)}))_{(0)}\cdot p_{(0)}\otimes m_{(0)})\otimes \mathscr{Q}^2(p_{(1)2}\otimes \zeta _{\gamma^{-1}} (n_{(1)})) \end{eqnarray*} $

and

$ \begin{eqnarray*} && c_{M\otimes N, P}(m\otimes n\otimes p)\\ &=&\mathscr{Q}(p_{(1)}\otimes \zeta _{\gamma^{-1}\delta\alpha^{-1}\delta^{-1}}((m\otimes n)_{\langle1\rangle}))(p_{(0)}\otimes (m\otimes n)_{\langle0\rangle})\\ &=& \mathscr{Q}(p_{(1)}\otimes \zeta _{\gamma^{-1}}(\zeta _{\delta \alpha^{-1}} (m_{(1)}) n_{(1)})) (p_{(0)}\otimes (m_{(0)}\otimes n_{(0)})). \end{eqnarray*} $

Take $M=N=P=A\otimes H$ and $m=1\otimes c$, and $n=1\otimes d$, and $p=1\otimes h$ for all $c, d, h\in H$. Then we obtain eq. (3.11).

Conversely, the proof is straightforward. We omit the details.

Lemma 3.9  For any $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$, $N \in {}_A{\cal {YD}}^{H}(\gamma, \delta)$ and $P \in {}_A{\cal {YD}}^{H}(\mu, \nu)$. Then eq. (3.7) holds if and only if the following condition is satisfied, with $\mathscr{U}=\mathscr{Q}$:

$ \begin{eqnarray} &&(\Delta^{cop}\otimes id)\mathscr{Q}(\zeta _{\mu }(d \zeta_{\gamma \mu ^{-1}}(h))) \otimes \zeta _{\alpha^{-1}}(c) = \mathscr{Q}^1(d\otimes \zeta _{\alpha^{-1}}(c_{2}))\otimes \mathscr{U}(h\otimes \zeta _{\alpha^{-1}} \nonumber\\ && (\mathscr{Q}^2(d\otimes \zeta _{\alpha^{-1}} (\zeta_{\beta} (\mathscr{Q}^2(d\otimes \zeta _{\alpha^{-1}} (c))_{(-1)}) m_{(1)1}\zeta_{\alpha}(S^{-1}\nonumber\\ && (\mathscr{Q}^2(d\otimes \zeta _{\alpha^{-1}} (c))_{(1)}) )))) (1\otimes \mathscr{Q}^2(d\otimes \zeta _{\alpha^{-1}} (c))_{(0)}) \end{eqnarray} $

(3.12)

for all $c, d, h\in H$.

Proof  If eq.(3.7) holds, then we have

$ \begin{eqnarray*} &&(id_{ {}^{M} N} \otimes c_{M, P} )\circ (c_{M, N}\otimes id_{P})(m\otimes n\otimes p)\\ &=&\mathscr{Q}^1(n_{(1)}\otimes \zeta _{\alpha^{-1}}(m_{(1)}))\cdot n_{(0)}\otimes \mathscr{U}(p_{(1)}\otimes \zeta _{\alpha^{-1}}[(\mathscr{Q}^2(n_{(1)} \otimes \zeta _{\alpha^{-1}}(m_{(1)}))\cdot m_{(0)})_{(1)}])\\ && (p_{(0)}\otimes (\mathscr{Q}^2(n_{(1)}\otimes \zeta _{\alpha^{-1}} (m_{(1)}))\cdot m_{(0)})_{(0)})\\ &=&\mathscr{Q}^1(n_{(1)}\otimes \zeta _{\alpha^{-1}}(m_{(1)2}))\cdot n_{(0)}\otimes \mathscr{U}(p_{(1)}\otimes \zeta _{\alpha^{-1}}(\mathscr{Q}^2(n_{(1)}\otimes \zeta _{\alpha^{-1}} (\zeta_{\beta} (\mathscr{Q}^2(n_{(1)}\\ &&\otimes \zeta _{\alpha^{-1}} (m_{(1)}))_{(-1)}) m_{(1)1}\zeta_{\alpha}(S^{-1})(\mathscr{Q}^2(n_{(1)}\otimes \zeta _{\alpha^{-1}} (m_{(1)}))_{(1)}) )))) (p_{(0)}\otimes \mathscr{Q}^2(n_{(1)}\\ && \otimes \zeta _{\alpha^{-1}} (m_{(1)}))_{(0)}\cdot m_{(0)}) \end{eqnarray*} $

and

$ \begin{eqnarray*} c_{M, N\otimes P}(m\otimes n\otimes p)=\mathscr{Q}(\zeta _{\mu }( n_{(1)}) \zeta _{\mu \gamma \mu ^{-1}}(p_{(1)})\otimes \zeta _{\alpha ^{-1}}(m_{(1)}))(n_{(0)}\otimes p_{(0)}\otimes m_{(0)}). \end{eqnarray*} $

Take $M=N=P=A\otimes H$ and $m=1\otimes c$, and $n=1\otimes d$, and $p=1\otimes h$ for all $c, d, h\in H$. Then we obtain eq. (3.12).

Conversely, it is straightforward.

Lemma 3.10  For any $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$, $N \in {}_A{\cal {YD}}^{H}(\gamma, \delta)$ and $P \in {}_A{\cal {YD}}^{H}(\mu, \nu)$. Then eq. (3.8) holds if and only if the following condition holds:

$ \begin{equation} (\xi _{\mu^{-1}\nu} \otimes \xi _{\mu^{-1}\nu }) \mathscr{Q}(\zeta _{\nu ^{-1}\delta \mu \delta ^{-1}}(d) \otimes \zeta _{\nu ^{-1}\mu \alpha ^{-1}}(c))=\mathscr{Q}(d \otimes \zeta _{\alpha^{-1}}(c)) \end{equation} $

(3.18)

for all $c, d\in H$.

Proof  Straightforward.

Therefore, we can summarize our results as follows.

Theorem 3.11  Let $A$ and $H$ be bialgebras and $\pi $ a group with the unit $e$. Let $\xi : \pi \to {\rm{Aut}}(A)$ and $\zeta : \pi \to {\rm{Aut}}(H)$ be group homomorphisms. Let $A$ be an $H$-bicomodule algebra and $\mathscr{Q}: H\otimes H \to A\otimes A$ a twisted convolution invertible map. Then the family of maps given by eq. (3.5) defines a braiding on the category ${}_A{\cal {YD}}^{H}_{G}$ if and only if equations (3.2)-(3.4) and (3.9)-(3.13) are satisfied.

Definition 3.12  Let $A$ and $H$ be bialgebras and $\pi $ a group with the unit $e$. Let $\xi: \pi \to {\rm{Aut}}(A)$ and $\zeta: \pi \to {\rm{Aut}}(H)$ be group homomorphisms. Let $A$ be an $H$-bicomodule algebra. We say that a $G$-double structure is $(A, H )$ together with a linear map $\mathscr{Q}: H\otimes H \to A\otimes A$ such that the following conditions hold:

(1) $\varepsilon _A(a)1_H=\varepsilon _A(a_{(0)})\zeta_\alpha(a_{(-1)})\zeta_\beta(S^{-1}(a_{(1)}));$

(2) $ a_{2(0)}\otimes a_{1(0)}\otimes \zeta_{\delta \beta}(a_{2(1)})\zeta_{\delta}(c)\zeta_{\delta\alpha}(S^{-1}(a_{2(-1)})) $$ \zeta_{\delta \alpha}(a_{1(1)}) \zeta_{\delta \alpha\delta^{-1}}(d) \zeta_{\delta \alpha\delta^{-1}\gamma}(S^{-1}(a_{1(-1)})))$$ =a_{(0)2}\otimes a_{(0)1} \otimes \zeta_{\delta\beta}(a_{(1)})\zeta_{\delta}(c)\zeta_{\delta\alpha\delta^{-1}}(d)\zeta_{\delta\alpha\delta^{-1}\gamma}(S^{-1}(a_{-1}));$

(3) $ \xi _{\alpha^{-1}\beta}(a)_{(0)}\otimes \zeta _{\beta^{-1}\delta\alpha\delta^{-1}}( \zeta _{\delta}(\xi _{\alpha^{-1}\beta}(a)_{(-1)}) c\zeta_{\gamma} (S^{-1}(\xi _{\alpha^{-1}\beta}(a)_{(-1)}))) $ $ = \xi _{\alpha^{-1}\beta }(a_{(0)})\otimes \zeta_{\beta^{-1}\delta\beta}(a_{(-1)}) \xi _{\beta^{-1}\delta\alpha\delta^{-1}}(c)\zeta_{\beta^{-1}\delta\alpha\delta^{-1}\gamma\delta^{-1}\beta}(S^{-1}(a_{(1)})); $

(4) $\mathscr{Q}(\zeta_{\delta}(a_{1(-1)})d\zeta_{\gamma}(S^{-1}(a_{1(1)}))\otimes \zeta _{\alpha^{-1}}(\zeta_{\beta}(a_{2(-1)})c\zeta_{\alpha}(S^{-1}(a_{2(1)}))) (a_{1(0)}\otimes a_{2(0)})$$=[(\xi _{\alpha^{-1}\beta} \otimes 1)\Delta ^{cop}(a)]\mathscr{Q} (d\otimes \zeta _{\alpha^{-1}}(c)); $

(5) $ \mathscr{Q}(d_1\otimes \zeta _{\alpha^{-1}}(c_1)) \otimes \zeta _{\delta }(c_2)\zeta _{\delta \alpha \delta^{-1}}(d_2)=\mathscr{Q}^1(d_2\otimes \zeta _{\alpha^{-1}}(c_2))_{(0)}\otimes \mathscr{Q}^2(d_2\otimes \zeta _{\alpha^{-1}}(c_2))_{(0)}$ $ \otimes \zeta _{\delta \alpha \delta^{-1}}(\zeta_{\delta} (\mathscr{Q}^1(d_{2}\otimes \zeta _{\alpha^{-1}}(c_{2}))_{(-1)})d_{1}\zeta_{\gamma}(S^{-1}(\mathscr{Q}^1(d_{2}\otimes \zeta _{\alpha^{-1}}(c_{2}))_{(1)})))$ $ \otimes \zeta _{\delta\alpha\delta^{-1}\gamma\alpha^{-1}}(\zeta_{\alpha} (\mathscr{Q}^2(d_{2}\otimes \zeta _{\alpha^{-1}}(c_{2}))_{(-1)})c_{1}\zeta_{\alpha}(S^{-1}(\mathscr{Q}^2(d_{2}\otimes \zeta _{\alpha^{-1}}(c_{2}))_{(1)})));$

(6) $ (id\otimes \Delta^{cop})\mathscr{Q}(h\otimes \zeta _{\gamma^{-1}}(\zeta _{\delta \alpha^{-1}} (c)d))=[(\xi _{\gamma ^{-1} \delta } \otimes 1)\mathscr{U}( \zeta_{\delta^{-1}\nu \delta}(\mathscr{Q}^1(h_{2}\otimes \zeta _{\gamma^{-1}} (d))_{(-1)}) $ $ h_{1} \zeta_{\delta^{-1}\nu \gamma \nu ^{-1}\mu \gamma ^{-1}\delta}(S^{-1}(\mathscr{Q}^1(h_{2}\otimes \zeta _{\gamma^{-1}} (d))_{(1)}))\otimes \zeta _{\alpha ^{-1}}(c))]$ $ (\mathscr{Q}^1(p_{(1)2}\otimes \zeta _{\gamma^{-1}} (d))_{(0)}\otimes 1)\otimes \mathscr{Q}^2(p_{(1)2}\otimes \zeta _{\gamma^{-1}} (d));$

(7) $ (\Delta^{cop}\otimes id)\mathscr{Q}(\zeta _{\mu }(d \zeta_{\gamma \mu ^{-1}}(e))) \otimes \zeta _{\alpha^{-1}}(c) = \mathscr{Q}^1(d\otimes \zeta _{\alpha^{-1}}(c_{2}))\otimes \mathscr{U}(h\otimes \zeta _{\alpha^{-1}} $ $ (\mathscr{Q}^2(d\otimes \zeta _{\alpha^{-1}} (\zeta_{\beta} (\mathscr{Q}^2(d\otimes \zeta _{\alpha^{-1}} (c))_{(-1)}) m_{(1)1}\zeta_{\alpha}(S^{-1}$ $ (\mathscr{Q}^2(d\otimes \zeta _{\alpha^{-1}} (c))_{(1)}) )))) (1\otimes \mathscr{Q}^2(d\otimes \zeta _{\alpha^{-1}} (m_{(1)}))_{(0)}\cdot m_{(0)});$

(8) $ (\xi _{\mu^{-1}\nu} \otimes \xi _{\mu^{-1}\nu }) \mathscr{Q}(\zeta _{\nu ^{-1}\delta \mu \delta ^{-1}}(d) \otimes \zeta _{\nu ^{-1}\mu \alpha ^{-1}}(c))=\mathscr{Q}(d \otimes \zeta _{\alpha^{-1}}(c));$

(9) ${\mbox{There exists a map:}}\, \, \mathscr{R}: H\otimes H \to A\otimes A\, \, \mbox{such that }$

$ \mathscr{Q}*\mathscr{R}(c\otimes d)=\mathscr{R}*\mathscr{Q}(c\otimes d)=\varepsilon(c)\varepsilon(d)1\otimes 1. $

Proposition 3.13  Let $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$ and assume that $M$ is finite-dimensional. Then

(1) If the following condition holds:

$ \begin{eqnarray} &&S^{-1}(a_{(0)})_{(0)}\otimes \zeta_{\beta ^{-1}}(a_{(-1)}) \zeta _{\beta^{-1}\alpha ^{-1}}S( \zeta_{\beta} (S^{-1}(a_{(0)})_{(-1)})h \zeta_{\alpha} S^{-1}\nonumber\\ &&((S^{-1}(a_{(0)})_{(1)})))\zeta_{\beta^{-1}\alpha ^{-1}\beta}(S^{-1}(a_{(-1)})) =S^{-1}(a)\otimes \zeta _{\beta^{-1}\alpha^{-1}}S(h) \end{eqnarray} $

(3.14)

for all $a\in A$ and $h\in H$, then ${M^*}$ is an object in $M \in {}_A{\cal {YD}}^{H}(\beta^{-1}\alpha^{-1}\beta, \beta ^{-1})$, with the module action and comodule coaction as follows:

$ \begin{eqnarray*} && (a\bullet f)(m)=f(S^{-1}(h)\cdot m), \\ && \rho (f)(m)=f_{\langle0\rangle}(m)\otimes f_{\langle1\rangle}=f(m_{(0)})\otimes \zeta _{\beta^{-1}\alpha ^{-1}}S(m_{(1)}) \end{eqnarray*} $

for $a\in A, f\in M^*$ and $m\in M$.

(2) The maps $b_{M}:k\rightarrow M\otimes {M^*}, \, \, b_{M}(1)=\sum_{i}e_{i}\otimes e^{i}$ (where $e_{i}$ and $e^{i}$ are dual bases in $M$ and ${M^*}$) and $d_{M}: M^{*}\otimes M \rightarrow k, \, \, d_{M}(f\otimes m)=f(m)$ are morphisms in ${}_A{\cal {YD}}^{H}_{G}$ and we have

$ ( id_{M}\otimes d_{M})(b_{M}\otimes id_{M})=id_{M}; \quad ( d_{M}\otimes id_{M^{*}})(id_{M^{*}}\otimes b_{M})=id_{M^{*}}. $

Proof  (1) For all $a\in A$ and $f\in M^*$, we compute

$ \begin{eqnarray*} &&(a_{(0)}\bullet f_{\langle0\rangle})(m)\otimes \zeta_{\beta ^{-1}}(a_{(-1)}) f_{\langle1\rangle}\zeta_{\beta^{-1}\alpha ^{-1}\beta}(S^{-1}(a_{(-1)}))\\ &=& f_{\langle0\rangle}(S^{-1}(a_{(0)})\cdot m)\otimes \zeta_{\beta ^{-1}}(a_{(-1)}) f_{\langle1\rangle}\zeta_{\beta^{-1}\alpha ^{-1}\beta}(S^{-1}(a_{(-1)}))\\ &=& f((S^{-1}(a_{(0)})\cdot m)_{(0)})\otimes \zeta_{\beta ^{-1}}(a_{(-1)}) \zeta _{\beta^{-1}\alpha ^{-1}}S((S^{-1}(a_{(0)})\cdot m)_{(1)})\zeta_{\beta^{-1}\alpha ^{-1}\beta}(S^{-1}(a_{(-1)}))\\ &=& f(S^{-1}(a_{(0)})_{(0)}\cdot m_{(0)}))\otimes \zeta_{\beta ^{-1}}(a_{(-1)}) \zeta _{\beta^{-1}\alpha ^{-1}}S( \zeta_{\beta} (S^{-1}(a_{(0)})_{(-1)})m_{(1)}\\ && \zeta_{\alpha} S^{-1}((S^{-1}(a_{(0)})_{(1)})))\zeta_{\beta^{-1}\alpha ^{-1}\beta}(S^{-1}(a_{(-1)}))\\ &\stackrel {(3.14)}{=}&f(S^{-1}(a)\cdot m_{(0)})\otimes \zeta _{\beta^{-1}\alpha^{-1}}S(m_{(1)})\\ &=&(a\bullet f)_{\langle0\rangle}(m)\otimes (a\bullet f)_{\langle 1\rangle} \end{eqnarray*} $

and as required.

(2) Straightforward.

Similarly, one has the following result.

Proposition 3.14  Let $M \in {}_A{\cal {YD}}^{H}(\alpha, \beta)$ and assume that $M$ is finite dimensional. Then

(1) If the following condition holds:

$ \begin{eqnarray} &&S(a_{(0)})_{(0)}\otimes \zeta_{\beta ^{-1}}(a_{(-1)}) \zeta _{\beta^{-1}\alpha ^{-1}}S^{-1}( \zeta_{\beta} (S(a_{(0)})_{(-1)})h \zeta_{\alpha} S^{-1}\nonumber\\ &&((S(a_{(0)})_{(1)})))\zeta_{\beta^{-1}\alpha ^{-1}\beta}(S(a_{(-1)})) =S(a)\otimes \zeta _{\beta^{-1}\alpha^{-1}}S^{-1}(h) \end{eqnarray} $

(3.15)

for all $a\in A$ and $h\in H$, then $^{*}M$ is an object in $M \in {}_A{\cal {YD}}^{H}(\beta^{-1}\alpha^{-1}\beta, \beta ^{-1})$, with the module action and comodule coaction as follows:

$ \begin{eqnarray*} && (a\bullet f)(m)=f(S(h)\cdot m), \\ && \rho (f)(m)=f_{\langle0\rangle}(m)\otimes f_{\langle1\rangle}=f(m_{(0)})\otimes \zeta _{\beta^{-1}\alpha ^{-1}}S^{-1}(m_{(1)}) \end{eqnarray*} $

for $a\in A, f\in M^*$ and $m\in M$.

(2) The maps $b_{M}:k\rightarrow ^{*}M\otimes M, \, \, b_{M}(1) =\sum\limits_{i}e^{i}\otimes e_{i}$ (where $e_{i}$ and $e^{i}$ are dual bases in $M$ and $^{*}M$) and $d_{M}:M\otimes {}^{*}M\rightarrow k, \, \, d_{M}(m\otimes f)=f(m)$ are morphisms in ${}_A{\cal {YD}}^{H}_{G}$ and we have

$ (d_{M}\otimes id_{M})(id_{M}\otimes b_{M})=id_{M}; \quad (id_{ \, \, ^{*}M}\otimes d_{M})(b_{M}\otimes id_{\, \, {}^{*}M})=id_{\, \, {}^{*}M}. $

Now, we consider ${}_A{\cal {YD}}^{H}_{G;fd}$, the subcategory of ${}_A{\cal {YD}}^{H}_{G}$ consisting of finite dimensional objects, then by Proposition 3.13 and Proposition 3.14, we get

Theorem 3.15  If equations (3.14) and (3.15) hold then ${}_A{\cal{YD}}^{H}_{G;fd}$ is a braided $T$-category with left and right dualities being given as in Proposition 3.13 and Proposition 3.14, respectively.

In this section we construct a quasitriangular $T$-coalgebra $\{ A\# H^*(\alpha, \beta )\}_{(\alpha, \beta )\in G}$, such that $\{{}_{A}\mathcal {YD}^{H}(\alpha, \beta)\}_{(\alpha, \beta )\in G}$ is isomorphic to the representation category of the quasitriangular $T$-coalgebra $\{ A\# H^*(\alpha, \beta )\}_{(\alpha, \beta )\in G}$.

Theorem 4.1  Let $G $ be a twisted semi-direct square group and $\mathscr{Q}: H\otimes H \to A\otimes A$ a linear map. Let $\xi : \pi \to Aut(A)$ and $\zeta : \pi \to Aut(H)$ be group homomorphisms. Let $(A, H, \mathscr{Q})$ be a $G$-double structure and assume $H$ is finite-dimensional with a dual basis $(e_i)_i\in H$ and $(e^i)_i\in H^*$. Then $A\# H^*=\{ A\# H^*(\alpha, \beta )\}_{(\alpha, \beta )\in G}$ is a $T$-coalgebra with the following structures:

The multiplication $m_{(\alpha, \beta )}$ and the unit of $A\# H^*(\alpha, \beta )$ are given, for any $a, b\in A$ and $h^*, g^*\in H^*$, by

$ (a\#{{h}^{*}})(b\#{{g}^{*}})=\sum\limits_{i}{\left\langle \left. {{h}^{*}}, {{\zeta }_{\beta }}({{b}_{(-1)}}){{e}_{i}}{{\zeta }_{\alpha }}({{S}^{-1}}({{b}_{(1)}})) \right\rangle \right.}a{{b}_{(0)}}\#{{e}^{i}}{{g}^{*}}, $

(4.1)

$ \begin{eqnarray} && 1_{A\# H^*(\alpha, \beta )}=1_A\otimes \varepsilon _H. \end{eqnarray} $

(4.2)

The comultiplication and the counit of $ A\# H^*$ are given by

$ \begin{eqnarray} &&\Delta_{(\alpha, \beta), (\gamma, \delta)}: A\# H^*((\alpha, \beta)\#(\gamma, \delta)) \to A\# H^*(\alpha, \beta)\otimes A\# H^*(\gamma, \delta), \nonumber\\ && \Delta_{(\alpha, \beta), (\gamma, \delta)}(a\#h^*)=(a_2\# \zeta ^*_{\delta ^{-1}}(h^*_1)\otimes (a_1\# \zeta ^*_{\delta\alpha ^{-1}\delta ^{-1}}(h_2^*)), \end{eqnarray} $

(4.3)

$\begin{eqnarray} && \varepsilon _{ A\# H^*}: A\# H^* \to k, \quad \varepsilon _{ A\# H^*}(a\# h^*)=(\varepsilon _A\otimes 1_H)(a\# h^*) \end{eqnarray} $

(4.4)

for all $a\in A$ and $h^*\in H^*$.

The antipode $S^{ A\# H^*}=\{S^{ A\# H^*}_{(\alpha, \beta )}: A\# H^*(\alpha, \beta ) \to A\# H^*((\alpha, \beta)^{-1})\}_{(\alpha, \beta)\in G}$ is given by

$\begin{align} & S_{(\alpha, \beta )}^{A\#{{H}^{*}}}(a\#{{h}^{*}})=\sum\limits_{i}{\left\langle \zeta _{{{\beta }^{-1}}{{\alpha }^{-1}}}^{*}({{S}^{*}}({{h}^{*}})), {{\zeta }_{{{\beta }^{-1}}}}({{S}^{-1}}{{(a)}_{(-1)}}){{e}_{i}}, \right.} \\ & {{\zeta }_{{{\beta }^{-1}}{{\alpha }^{-1}}\beta }}({{S}^{-1}}({{S}^{-1}}{{(a)}_{(1)}}))>{{S}^{-1}}{{(a)}_{(0)}}\#{{e}^{i}}. \\ \end{align} $

(4.5)

The crossing $\varphi =\{\varphi ^{(\gamma, \delta)}_{(\alpha, \beta )}: A\# H^*(\gamma, \delta ) \to A\# H^*((\alpha, \beta)\#(\gamma, \delta )\#(\alpha, \beta )^{-1})\}$ is defined by

$ \begin{equation} \varphi ^{(\gamma, \delta)}_{(\alpha \beta )}(a\#c^*)=\xi _{\beta^{-1}\alpha }(a)\# \zeta ^*_{\beta ^{-1}\delta \alpha \delta ^{-1}}(h^*). \end{equation} $

(4.6)

Proof  First, the multiplication is associative and the unit is $1_A\otimes \varepsilon _H$.

Second, it is straightforward to check that $\varphi $ satisfies equation (2.10), (2.11) and (2.12), i.e., the following conditions hold:

$\varphi $ is multiplicative, i.e., $\varphi _{(\alpha, \beta )}\circ \varphi _{(\gamma, \delta)}=\varphi _{(\alpha, \beta )\#(\gamma, \delta )}$, in particular $\varphi ^{(\gamma, \delta )}_{(e, e)}=id$.

$\varphi $ is compatible with $\Delta $, i.e.,

$ \Delta_{(\mu, \nu)\#(\alpha, \beta )\#(\mu, \nu)^{-1}, \, (\mu, \nu)\#(\gamma , \delta )\#(\mu, \nu)^{-1}}\circ \varphi ^{(\alpha, \beta )\#(\gamma, \delta)}_{(\mu, \nu)}= (\varphi ^{(\alpha, \beta )}_{(\mu, \nu)}\otimes \varphi ^{(\alpha, \beta )}_{(\mu, \nu)})\circ \Delta_{(\alpha, \beta ), \, (\gamma, \delta )}. $

$\varphi $ is compatible with $\varepsilon $, i.e., $\varepsilon \circ \varphi ^{(e, e)}_{(\alpha, \beta )}=\varepsilon $ for any $(\alpha, \beta )\in G$.

Third, the coassociativity follows directly from the coassociativity of the comultiplication of $A$ and $H^*$ and the fact $\varphi _{(\alpha, \beta )}\circ \varphi _{(\gamma, \delta)}=\varphi _{(\alpha, \beta )\#(\gamma, \delta )}$. It is easy to check that $\varepsilon _{A\# H^*}$ is multiplicative.

Fourth, we show that $\Delta_{(\alpha, \beta), (\gamma, \delta)}$ is an algebra morphism, i.e., axiom (2.8) is satisfied. For any $a, b\in A$ and $h^*, g^*\in H^*$, we do calculations as follows:

$ \begin{eqnarray*} &&\Delta_{(\alpha, \beta), (\gamma, \delta)}[(a\# h^*)(b\# g^*)]\\ &=&\left\langle \left. { h^*, e_i^{\psi ((\alpha, \beta )\# (\gamma, \delta )}} \right\rangle \right. (a {}_{\psi }b)_2\# \zeta ^*_{\delta ^{-1}}((e^id^*)_1)\otimes (a {}_{\psi }b)_1\# \zeta ^*_{\delta \alpha ^{-1}\delta ^{-1}}((e^id^*)_2)\\ &=&\left\langle \left. { h^*, \zeta_{\delta\beta}(b_{(-1)})e_j e_i\zeta_{\delta\alpha\delta^{-1}\gamma}(S^{-1}(b_{(1)}))} \right\rangle \right. a_2 (b_{(0)})_2\# \zeta ^*_{\delta ^{-1}}(e^jg^*_1)\otimes a_1 (b_{(0)})_1\# \zeta ^*_{\delta \alpha ^{-1}\delta ^{-1}}(e^i g^*_2)\\ &=&\left\langle \left. { h^*, \zeta_{\delta\beta}(b_{(-1)})\zeta _{\delta}(e_j)\zeta _{\delta \alpha \delta ^{-1}}(e_i)\zeta_{\delta\alpha\delta^{-1}\gamma}(S^{-1}(b_{(1)}))} \right\rangle \right. a_2 (b_{(0)})_2\\ &&\# e^j\zeta ^*_{\delta ^{-1}}(g^*_1)\otimes a_1 (b_{(0)})_1\# e^i\zeta ^*_{\delta \alpha ^{-1}\delta ^{-1}}( g^*_2)\\ &=&\left\langle \left. { \zeta ^*_{\delta ^{-1}}(h^*), \, \, \zeta_{\beta}((b_2)_{(-1)})e_j\zeta_{\alpha}(S^{-1}((b_2)_{(1)})) \zeta _{\alpha \delta ^{-1}}(\zeta_{\delta}((b_1)_{(-1)})e_i\zeta_{\gamma}(S^{-1}((b_1)_{(1)})))} \right\rangle \right.\\ &&a_2 \, (b_2)_{(0)}\# e^j \zeta ^*_{\delta ^{-1}}(d^*_1)\otimes a_1\, (b_1)_{(0)}\# e^i \zeta _{\delta \alpha ^{-1} \delta ^{-1}}(g^*_2)\\ &=&(a_2\# \zeta ^*_{\delta ^{-1}}(h^*_1)(b_2\# \zeta ^*_{\delta ^{-1}}(g^*_1) \otimes (a_1\# \zeta ^*_{\delta \alpha ^{-1}\delta ^{-1}}(h^*_2) (b_1\# \zeta ^*_{\delta \alpha ^{-1}\delta ^{-1}}(d^*_2)\\ &=&\Delta_{(\alpha, \beta)}(a\# h^*)\Delta_{(\gamma, \delta)}(b\# g^*). \end{eqnarray*} $

Finally, for all $(\alpha, \beta)\in G$, we have to check axiom (2.9). We now prove one of them as follows:

$ \begin{eqnarray*} &&m_{(\alpha, \beta )^{-1}}\circ (S^{ A\# H^*}_{(\alpha, \beta)}\otimes id_{(\alpha, \beta )^{-1}})\circ \Delta _{(\alpha, \beta ), (\alpha, \beta )^{-1}}(a\# h^*)\\ &=&S^{ A\# H^*}_{(\alpha, \beta)}(a_2\# \zeta ^*_{\beta}(h_1^*)(a_1\# \zeta ^*_{\beta^{-1} \alpha^{-1} \beta}(h^*_2)\\ &=&\sum _i\left\langle \left. {\zeta ^*_{\beta ^{-1}\alpha ^{-1}}\zeta ^*_{\beta }S^*(h_1^*), \zeta_{\beta^{-1}}(S^{-1}(a_2)_{(-1)})e_i \zeta_{\beta^{-1}\alpha^{-1}\beta}(S^{-1}(S^{-1}(a_2)_{(1)}))} \right\rangle \right.\\ &&(S^{-1}(a_2)_{(0)}\# e^i)(a_1\# \zeta ^*_{\beta^{-1} \alpha^{-1} \beta}(h^*_2)\\ &\stackrel {(4.1)}{=}&\sum _{i, j}\left\langle \left. {\zeta ^*_{\beta^{-1} \alpha ^{-1} \beta }S^*(h_1^*), \zeta_{\beta^{-1}}(S^{-1}(a_2)_{(-1)})e_i \zeta_{\beta^{-1}\alpha^{-1}\beta}(S^{-1}(S^{-1}(a_2)_{(1)})} \right\rangle \right. \\ &&\left\langle \left. {e^i, \zeta_{\beta^{-1}}(a_{1(-1)})e_j \zeta_{\beta^{-1}\alpha^{-1}\beta}(S^{-1}(a_{1(1)})} \right\rangle \right. (S^{-1}(a_2)_{(0)}a_{1(0)} \# e^j \zeta ^*_{\beta^{-1} \alpha^{-1} \beta}(h^*_2))\\ &=&\sum _{i, j}\left\langle \left. {\zeta ^*_{\beta^{-1} \alpha ^{-1} \beta }S^*(h_1^*), \zeta_{\beta^{-1}}((S^{-1}(a_2)a_1)_{(-1)}) e_i \zeta_{\beta^{-1}\alpha^{-1}\beta}(S^{-1}((S^{-1}(a_2)a_1)_{(1)}))} \right\rangle \right.\\ &&((S^{-1}(a_2)a_1)_{(0)}\# e^j\zeta ^*_{\beta^{-1} \alpha^{-1} \beta}(h^*_2))\\ &=&\varepsilon _A(a)\zeta ^*_{\beta^{-1} \alpha ^{-1} \beta }S^*(h_1^*)\zeta ^*_{\beta^{-1} \alpha^{-1} \beta}(h^*_2))\\ &=&1_{(\alpha, \beta )}\varepsilon_{A\otimes H^*}(a\# h^*), \end{eqnarray*} $

and the other one can be verified in the similar way.

Theorem 4.2  Let $G $ be a twisted semi-direct square group and and $\mathscr{Q}: H\otimes H \to A\otimes A$ a linear map. Let $\xi : \pi \to {\rm{Aut}}(A)$ and $\zeta : \pi \to {\rm{Aut}}(H)$ be group homomorphisms. Let $(A, H, \mathscr{Q})$ be a $G$-double structure and $H$ a finite-dimensional with a dual basis $(e_i)_i\in H$ and $(e^i)_i\in H^*$. Then the category ${}_A{\cal {YD}}^{H}$ is isomorphic to the category Rep $(A\# H^*)$ of representations of $A\# H^*$ as braided $T$-categories. Moreover, $A\# H^*=\{ A\# H^*(\alpha, \beta )\}_{(\alpha, \beta ) \in G }$ is a quasitriangular $T$-coalgebra with the quasitriangular structure given by

$ \begin{eqnarray*} && R =\{R_{(\alpha, \beta), (\gamma, \delta ) }\\ &=&\sum _{i, j} e^i\# \mathscr{Q}^1(e_i\otimes \zeta _{\alpha }(e_j)) \otimes e^j\# \mathscr{Q}^2(e_i\otimes \zeta _{\alpha }(e_j)) \in A\# H^*(\alpha, \beta)\otimes A\# H^*(\gamma, \delta)\} \end{eqnarray*} $

for all $\alpha, \beta, \gamma, \delta \in \pi$.

Proof  Since $(A, H, \mathscr{Q})$ is a $G$-double structure we have the braided $T$-category ${}_A{\cal {YD}}^{H}_{G}$. The braiding on ${}_A{\cal {YD}}^{H}_{G}$ translates into a braiding on the category $Rep(A\# H^*)$ of representations of $A\# H^*$. But this means that $A\# H^*=\{ A\# H^*(\alpha, \beta )\}_{(\alpha, \beta )\in G}$ is a quasitriangular $T$-coalgebra. The invertible map $\mathscr{Q}: H\otimes H \to A\otimes A$ satisfying the conditions (4), (5), (6) and (7) in Definition 3.12 induces a map

$ \widetilde{\mathscr{Q}}: k \to A\# H^*(\alpha, \beta)\otimes A\# H^*(\gamma, \delta). $

Then $\widetilde{\mathscr{Q}}(1)$ is just the corresponding $R_{(\alpha, \beta), (\gamma, \delta )}\in A\# H^*(\alpha, \beta)\otimes A\# H^*(\gamma, \delta)$.

In this case, we have the braiding on the category $Rep(A\# H^*)$:

$ c_{M, N}: M \otimes N \rightarrow {}^{M}N \otimes M, \, m\otimes n\mapsto [\tau _{(\gamma, \delta), (\alpha, \beta )} R_{(\alpha, \beta), (\gamma, \delta )}](n\otimes m) $

for any $M\in {}_{A\# H^*(\alpha, \beta)}{\mathscr M}$ and $N\in {}_{A\# H^*(\gamma, \delta )}{\mathscr M}$.