Braided $T$-categories introduced by Turaev [1] are of interest due to their applications in homotopy quantum field theories, which are generalizations of ordinary topological quantum field theories. Braided crossed categories based on a group $G$, is braided monoidal categories in Freyd-Yetter categories of crossed $G$-sets (see [2]) play a key role in the construction of these homotopy invariants. In [3], Zhou and Yang studied cotriangular weak Hopf group-coalgebras and promoted Kegel theorem on the weak Hopf group-coalgebras. Motivated by this fact, the author Yang [4] introduced the notion of a monoidal Hom-group-coalgebra as a development of the notion of monoidal Hom-coalgebras in sense of Caenepeel and Goyvaerts (see [5]), and as a natural generalization of the notions of both the Hom-type Hopf algebras and the Hopf group-coalgebra in [1, 6], and constructed a new kind of braided T-categories.

Starting from a finite-dimensional Hopf algebra $H$, Drinfeld [7] showed how to obtain a quasitriangular Hopf algebra $D(H)$, the quantum double of $H$. It is now very natural to ask how to construct Drinfeld quantum double for finite-type monoidal Hom-Hopf group-coalgebras. In this article, we essentially construct Drinfeld quantum double over monoidal Hom-Hopf group-coalgebras.

This article is organized as follows. In Section 1, we recall some notions and results about monoidal Hom-Hopf group-coalgebras. In Section 2, we construct the Drinfeld quantum double over monoidal Hom-Hopf group-coalgebras and study quasitriangular monoidal Hom-Hopf group-coalgebras.

In this section, we recall the definitions and properties of monoidal Hom-Hopf algebras and monoidal Hom-Hopf group-coalgebras. Throughout this paper, we always let $G$ be a discrete group with a neutral element $1$ and $k$ a field. If $U$ and $V$ are $k$-spaces, $T_{U, V}: U\otimes V\rightarrow V\otimes U$ will denote the flip map defined by $T_{U, V}(u\otimes v)=v\otimes u$ for all $u\in U$ and $v\in V$.

Definition 2.1 (see [4]) A monoidal Hom-$G$-coalgebra is a family of $k$-spaces $C=\{(C_{\alpha }, \xi_{C_{\alpha }})\}_{\alpha \in G}$ together with a family of $k$-linear maps $\Delta=\{ \Delta_{\alpha , \beta}: C_{\alpha \beta}\rightarrow C_{\alpha } \otimes C_{\beta}\}_{\alpha , \beta\in G}$ and a $k$-linear map $\varepsilon: C_1\rightarrow k$, such that $\Delta$ is coassociative in the sense that

$ ({{\Delta }_{\alpha ,\beta }}\otimes \xi _{{{C}_{\gamma }}}^{-1}){{\Delta }_{\alpha \beta ,\gamma }}=(\xi _{{{C}_{\alpha }}}^{-1}\otimes {{\Delta }_{\beta ,\gamma }}){{\Delta }_{\alpha ,\beta \gamma }}\text{ for any }\alpha ,\beta ,\gamma \in G, $

(2.1)

$ (\varepsilon\otimes \xi_{C_\alpha }) \Delta_{1, \alpha }=\xi_{C_\alpha }^{-1}=(\xi_{C_\alpha }\otimes \varepsilon)\Delta_{\alpha , 1} \mbox{for all } \alpha \in G. $

(2.2)

Remark 2.2 $(C_1, \xi_{C_{1}}, \Delta_{1, 1}, \varepsilon)$ is a monoidal Hom-coaglegbra in the sense of Caenepeel and Goyvaerts [5].

Following the Sweedler's notation for $G$-coalgebras, for any $\alpha , \beta \in G$ and $c\in (C_{\alpha \beta }, \xi_{C_{\alpha \beta }})$ one writes

$ \Delta_{\alpha , \beta }(c)=c_{(1, \alpha )}\otimes c_{(2, \beta )}\in C_{\alpha }\otimes C_{\beta }. $

The coassociativity axiom (2.1) gives that, for any $\alpha , \beta , \gamma \in G$ and $c\in (C_{\alpha \beta \gamma }, \xi_{C_{\alpha \beta \gamma }}), $

$ (c_{(1, \alpha \beta )(1, \alpha )}\otimes c_{(1, \alpha \beta )(2, \beta )})\otimes \xi_{C_\gamma }^{-1}(c_{(2, \gamma )}) = \xi_{C_\alpha }^{-1}( c_{(1, \alpha )})\otimes (c_{(2, \beta \gamma )(1, \beta )}\otimes c_{(2, \beta \gamma )(2, \gamma )}). $

(2.3)

Definition 2.3 (see [4]) A monoidal Hom-Hopf $G$-coaglebra is a monoidal Hom-$G$-coalgebra $H=(\{H_{\alpha }, \xi_{H_{\alpha }}\}, \Delta, \varepsilon)$ together with a family of $k$-linear maps $S=\{S_\alpha :H_\alpha \rightarrow H_{\alpha ^{-1}}\}_{\alpha \in G}$ such that the following data holds:

$ {\rm{Each}}\;{H_\alpha }\;{\rm{is}}\;{\rm{a}}\;{\rm{monoidal}}\;{\rm{Hom-algebra}}\;{\rm{with}}\;{\rm{multiplication}}\;{m_\alpha }\;{\rm{and}}\;{\rm{unit}}\;{1_\alpha } \in {H_\alpha }. $

(2.4)

$ {\rm{For}}\;{\rm{all}}\;\alpha , \beta \in G, {\Delta _{\alpha , \beta }}\;and\;\varepsilon :{H_1} \to k\;{\rm{are}}\;{\rm{algebra}}\;{\rm{maps}}. $

(2.5)

$ {\rm{For}}\;\alpha \in G, {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}}}}. $

(2.6)

Note that $(H_1, m_1, 1_1, \Delta_{1, 1}, \varepsilon, S_1)$ is a monoidal Hom-Hopf algebra. A monoidal Hom-Hopf $G$-coalgebra $H$ is termed to be of finite type if, for all $\alpha \in G$, $H_{\alpha }$ is finite-dimensional as $k$-vector space.

Remark 2.4 Let $H=(\{H_{\alpha }, \xi_{H_{\alpha }}\}, \Delta, \varepsilon, S)$ be a monoidal Hom-Hopf $G$-coalgebra. Suppose that the antipode $S=\{S_\alpha \}_{\alpha \in G}$ of $H$ is bijective. For any $\alpha \in G$, let $H_{\alpha }^{op}$ be the opposite algebra to $H_\alpha .$ Then $H^{op}=\{H_\alpha ^{op}\}_{\alpha \in G}$, endowed with the comultiplication and the counit of $H$ and with the antipode $S^{op}=\{S_\alpha ^{op}=S_{\alpha ^{-1}}^{-1}\}_{\alpha \in G}, $ is an opposite monoidal Hom-Hopf $G$-coalgebra of $H.$ The coopposite monoidal Hom-$G$-coalgebra equipped with $H_\alpha ^{cop}=H_{\alpha ^{-1}}$ as an algebra and with the comultiplication $\Delta_{\alpha , \beta }=T_{C_{\beta{-1}}, C_{\alpha ^{-1}}}\Delta_{\beta^{-1}, \alpha ^{-1}}$ and with the antipode $S^{cop}=\{S_{\alpha }^{cop}=S_{\alpha }^{-1}\}_{\alpha \in G}.$

Definition 2.5 (see [4]) A monoidal Hom-$G$-coalgebra $H=(\{H_{\alpha }, \xi_{H_{\alpha }}\}, \Delta, \varepsilon, S)$ is said to be a monoidal Hom-$T$-coalgebra provided it is endowed with a family of algebra isomorphisms $\varphi=\{\varphi_{\beta }:H_{\alpha }\rightarrow H_{\beta\alpha \beta ^{-1}}\}_{\alpha , \beta \in G}$ such that each $\varphi_\beta $ preserves the comultiplication and the counit, i.e., for all $\alpha , \beta , \gamma \in G, $

$ (\varphi_\beta \otimes \varphi_\beta )\circ \Delta_{\alpha , \gamma } =\Delta_{\beta\alpha \beta ^{-1}, \beta \gamma \beta ^{-1}}\circ\varphi_\beta , \varepsilon\circ\varphi_{\beta }=\varepsilon, $

and $\varphi$ is multiplicative in the sense that $\varphi_{\alpha \beta }=\varphi_\alpha \circ\varphi_\beta , $ for all $\alpha , \beta \in G.$

Let $H$ be a monoidal Hom-$T$-coalgebra. Then one has that $\varphi_1|H_{\alpha }=id_{H_\alpha }, $ $\varphi_{\alpha }^{-1}=\varphi_{\alpha ^{-1}}, $ for all $\alpha \in G$ and $\varphi$ preserves the antipode, i.e., $\varphi_{\beta}\circ S_{\alpha } =S_{\beta\alpha \beta^{-1}}\circ\varphi_{\beta}$ for all $\alpha , \beta\in G.$

In order to construct the Drinfeld quantum double for monoidal Hom-Hopf $T$-coalgebras and study the definition of quasitriangular monoidal Hom-Hopf group-algebra. The following definitions are necessary.

Definition 3.1 The Duality $C^*$. Let $C=(\{C_{\alpha }, \xi_{C_\alpha }, \Delta, \varepsilon\})$ be a $G$-coalgebra and $A$ an algebra with multiplication $m$ and unit element $1_A.$ For any $f\in {\rm Hom}_{k}(C_\alpha , A)$ and $g\in {\rm Hom}_{k}(C_\beta , A), $ we have their convolution product by

$ (f*g)(c)=m(f\otimes g)\Delta_{\alpha , \beta }(c)=f(c_{(1, \alpha )})g(c_{(2, \beta )})\in {\rm Hom}_{k}(C_{\alpha , \beta }, A) $

for all $c\in C_{\alpha , \beta }.$ Equations (2.1) and (2.2) will imply that $k$-space

$ \text{Conv}(C,A)=\underset{\alpha \in G}{\mathop \oplus }\,\text{Ho}{{\text{m}}_{k}}({{C}_{\alpha }},A) $

endowed with the convolution product $*$ and the unit element $1_{A}\varepsilon$, is a $G$-algebra, called a convolution algebra.

In particular, for $A=k$, the $G$-algebra $Conv(C, k)=\oplus_{\alpha \in G}C_{\alpha }^*$ is called dual to $C$ and is denoted by $C^*$.

Definition 3.2 The Mirror $\overline{H}$. Let $H$ be a monoidal Hom-$T$-coalgebra. Then the notion of the mirror $\overline{H}$ of $H$ is given by the following data.

$ \bullet $ For any $\alpha \in G, $ set $\overline{H}_\alpha =H_{\alpha ^{-1}}.$

$ \bullet $ For any $\alpha , \beta \in G, $ the $G$-coalgebra structure is defined by

$ \overline{\Delta}_{\alpha , \beta } =((\varphi_\beta \otimes id_{H_{\beta^{-1}}})\circ\Delta_{\beta^{-1}\alpha \beta , \beta ^{-1}})(h)\nonumber\\ =\varphi_\beta (h_{(1, \beta ^{-1}\alpha ^{-1}\beta)})\otimes h_{(2, \beta ^{-1})}\in \overline{H}_\alpha \otimes \overline{H}_\beta $

(3.1)

for any $h\in \overline{H_{\alpha \beta }}=H_{\beta^{-1}\alpha ^{-1}}.$ The counit of $\overline{H}$ is given by $\varepsilon\in H_{1}^*=\overline{H}_{1}^*.$

$ \bullet $ For any $\alpha \in G, $ the $\alpha $th component of the antipode $\overline{S}$ of $\overline{H}$ is given by $\overline{S}_{\alpha }=\varphi_\alpha \circ S_{\alpha ^{-1}}.$

$ \bullet $ For any $\alpha \in G, $ the $\alpha $th component of the crossed map $\overline{\varphi}$ of $\overline{H}$ is given by $\overline{\varphi}_{\alpha }=\varphi_\alpha .$

Dually, a monoidal Hom-$G$-algebra is a family of $k$-spaces $A=\{(A_{\alpha }, \xi_{A_{\alpha }})\}_{\alpha \in G}$ together with a family of $k$-linear maps $m=\{m_{\alpha , \beta }: A_{\alpha }\otimes A_{\beta }\rightarrow A_{\alpha \beta }\}_{\alpha , \beta \in G}$ and a $k$-linear map $\eta: k\rightarrow A_1$, such that $m$ is associative in the sense that, for any $\alpha , \beta , \gamma \in G, $

$ m_{\alpha \beta , \gamma }(m_{\alpha , \beta }\otimes \xi_{A_\gamma })= m_{\alpha , \beta \gamma }(\xi_{A_\alpha }\otimes m_{\beta, \gamma }), $

(3.2)

and for all $\alpha , \beta \in G, $

$ m_{\alpha , 1}(id_{A_\alpha }\otimes \eta)=\xi_{A_\alpha }=m_{1, \alpha }(\eta\otimes id_{A_\alpha }). $

(3.3)

A monoidal Hom-Hopf $G$-algebra is a $G$-algebra $H=(\{H_\alpha , \xi_{H_\alpha }\}, m, \eta)$ endowed with a family of $k$-linear maps $S=\{S_\alpha : H_\alpha \rightarrow H_{\alpha ^{-1}}\}_{\alpha \in G}$ such that each $(H_\alpha , \xi_{H_\alpha })$ is a monoidal Hom-coalgebra with a comultiplication $\Delta_\alpha $ and a counit $\varepsilon_\alpha $; the map $\eta: k\rightarrow A_1$ and the maps $m_{\alpha , \beta }:H_\alpha \otimes H_\beta \rightarrow H_{\alpha \beta }$ (for all $\alpha , \beta \in G$) are coalgebra homomorphisms; and for any $\alpha \in G, $ one has that

$ m_{\alpha ^{-1}, \alpha }(S_{\alpha }\otimes id_{H_{\alpha }})\Delta_{\alpha } =\varepsilon_\alpha 1_1 = m_{\alpha , \alpha ^{-1}}(id_{H_{\alpha }}\otimes S_{\alpha })\Delta_{\alpha }. $

(3.4)

A monoidal Hom-Hopf $G$-algebra $H$ is said to be of finite type if, for all $\alpha \in G, H_\alpha $ is finite dimensional as $k$-space.

Furthermore, a monoidal Hom-Hopf $T$-algebra is a monoidal Hom-Hopf $G$-algebra $H$ with a set of coalgebra isomorphisms $\psi= \{\psi_{\beta}: H_{\alpha }\rightarrow H_{\beta\alpha \beta^{-1}}\}_{\alpha , \beta\in G}$ called a conjugation, satisfying the following conditions:

$ \bullet $ $\psi$ is multiplicative, i.e., $\psi_{\beta}\circ\psi_{\gamma }=\psi_{\beta\gamma }\, $ for any $\beta, \gamma \in G.$ It follows that, for any $\alpha \in G, $ $\psi_{1}|H_{\alpha }=id_{H_\alpha }.$

$ \bullet $ $\psi$ is compatible with $m$, i.e., for any $\alpha , \beta, \gamma \in G, $ we have $m_{\gamma \alpha \gamma ^{-1}, \gamma \beta \gamma ^{-1}}\circ(\psi_\gamma \otimes \psi_\gamma )=\psi_\gamma \circ m_{\alpha , \beta }.$

$ \bullet $ $\psi$ is compatible with $\eta$, i.e., $\eta\circ\psi_{\gamma }=\eta$ for any $\gamma \in G.$

Let $H$ be a monoidal Hom-Hopf $T$-algebra. Similar to that of [9] we have the construction $H_{pk}$ (called a packed form of $H$) which can form a Hom-Hopf algebra.

Remark 3.3 Let $H$ be a finite type monoidal Hom-Hopf $T$-algebra. The dual of $H$ is the monoidal Hom-Hopf $T$-algebra defined as follows. For any $\alpha \in G, $ the $\alpha $th component of $H^{*}$ is the dual coalgebra $(H_{\alpha }^{*}, \xi_{\alpha }^{*-1})$ of the algebra $(H_{\alpha }, \xi_{\alpha }).$ The multiplication of $H^{*}$ is given by

$ \left\langle {{m_{\alpha , \beta }}(f \otimes g), h} \right\rangle = \left\langle {f \otimes g, {\Delta _{\alpha , \beta }}} \right\rangle $

(3.5)

for any $f\in (H_{\alpha }^{*}, \xi_{\alpha }^{*-1}), g\in (H_{\beta}^{*}, \xi_{\beta}^{*-1})$ and $h\in (H_{\alpha \beta}, \xi_{\alpha \beta}), $ with $\alpha , \beta\in G.$ The unit of $H^{*}$ is given by $\varepsilon\in H_{1}^{*}\subset H^{*}.$ The antipode $ \mathscr{A}^{*}$ of $H^{*}$ is given by $ \mathscr{A}_{\alpha }^{*}=S_{\alpha ^{-1}}^{*}, $ for any $\alpha \in G.$ For any $\beta\in G, $ the conjugation isomorphism $\psi_{\beta}^{*}=\varphi_{\beta^{-1}}^{*}.$

Remark 3.4 Given any crossed monoidal Hom-Hopf $T$-coalgebra, then $((H^{*})_{pk})^{cop}$ is the monoidal Hom-Hopf algebra obtained from $(H^{*})_{pk}$ by replacing its comultiplication with the new one $\Delta^{*}=\Delta^{*t, cop}$ given by

$ \left\langle {{\Delta ^*}(f), h \otimes k} \right\rangle = \left\langle {f, kh} \right\rangle $

(3.6)

for any $f\in H_{\alpha }^{*}\subset \oplus _{\beta\in G} H_{\beta}^{*}$ and $h, k\in H_{\alpha }, $ with $\alpha \in G.$ We also replace the antipode with the new one obtained by $\mathscr{A}_{*}=\mathscr{A}^{*t}=(S^*)^{-1}.$ In particular, we have $ < \mathscr{A}_{*}(f), h>= < f, S_{\alpha }^{-1}(h)>, $ for any $f \in H_{\alpha }^*$ and $h\in H_{\alpha ^{-1}}, $ with $\alpha \in G.$ We can obtain the crossed monoidal Hom-Hopf $T$-coalgebra denoted by $H^{*t, cop}$ based on $((H^{*})_{pk})^{cop}.$ Note that $\varphi_{H^{*t, cop}, \alpha }=\varphi_{H^{*t}, \alpha } =\sum\limits_{\beta\in G}\varphi_{\beta^{-1}}^{*}, $ for any $\alpha \in G.$

Let $H$ be a finite type monoidal Hom-Hopf $T$-coalgebra. We define the Drinfeld quantum double $D(T)$ of $H$ as follows. Consider the following vector spaces

$ {{H}_{{{\alpha }^{-1}}}}\otimes H_{\alpha }^{*t,cop}={{H}_{{{\alpha }^{-1}}}}\otimes H_{1}^{*t,cop}={{{\bar{H}}}_{\alpha }}\otimes \underset{\beta \in G}{\mathop \oplus }\,H_{\beta }^{*} $

for any $\alpha \in G.$ A multiplication is obtained by setting, for any $h, k\in H_{\alpha ^{-1}}, f\in H_{\gamma }^{*}, $ and $g\in H_{\delta}^*, $ with $\gamma , \delta\in G, $

$ \begin{align} &(h\circledast f)(k\circledast g) \\ &=\xi _{{{\alpha }^{-1}}}^{2}({{h}_{(12, {{\alpha }^{-1}})}})k\circledast f\left\langle g, S_{\delta }^{-1}({{h}_{(2, {{\delta }^{-1}})}})((\cdot ){{\varphi }_{\alpha }}({{h}_{(11, {{\alpha }^{-1}}\delta \alpha )}})) \right\rangle \\ &\text{=}\xi _{{{\alpha }^{-1}}}^{2}({{h}_{(12, {{\alpha }^{-1}})}})k\circledast f\left\langle {{g}_{11}}, {{\varphi }_{\alpha }}({{h}_{(11, {{\alpha }^{-1}}\delta \alpha )}}) \right\rangle \left\langle {{g}_{2}}, S_{\delta }^{-1}({{h}_{(2, {{\delta }^{-1}})}}) \right\rangle \xi _{\delta }^{*-2}({{g}_{12}}). \\ \end{align} $

(3.7)

For any $h, k\in H_{\alpha ^{-1}}$ and $f\in H_{\gamma }^{*}$

$ (h\circledast f)(k\circledast \varepsilon) =hk\circledast \xi_{\gamma }^{*-1}(f). $

We now have the following main result of this section.

Theorem 3.5 Let $H$ be a finite-type monoidal Hom-Hopf $T$-coalgebra. Then $D(H)$ is a crossed monoidal Hom-Hopf $T$-coalgebra with the following structures:

$ \bullet $ For any $\alpha \in G, \, \alpha $th component $D_{\alpha }(H)$ is an associative algebra with the multiplication given in eq. (3.7) and with unit $ 1_{\alpha ^{-1}}\circledast \varepsilon;$

$ \bullet $ The comultiplication is given by

$ \Delta_{\alpha , \beta }(h\circledast F)= [\varphi_{\beta }(h_{(1, \beta ^{-1}\alpha ^{-1}\beta)})\circledast F_{1}] \otimes [h_{(2, \beta ^{-1})}\circledast F_{2}] $

(3.8)

for any $\alpha , \beta \in G, h\in \bar{H}_{\alpha \beta }$ and $F\in H^{*t, cop}, $ where we have that $\Delta^{*}(F)=F_1\otimes F_2$ defined by eq. (3.6);

$ \bullet $ The counit is obtained by setting

$ \varepsilon (h\circledast f)=\left\langle \varepsilon, h\circledast f \right\rangle \text{=}\left\langle \varepsilon, h \right\rangle \left\langle f, {{1}_{\gamma }} \right\rangle $

(3.9)

for any $h\in H_{1}$ and $f\in H_{\gamma }^{*}, $ with $\gamma \in G;$

$ \bullet $ For any $\alpha \in G, $ the $\alpha $th component of the antipode of $D(H)$ is given by

$ \begin{align} &{{S}_{\alpha }}(h\circledast F)\text{=} \\ &[{{{\bar{S}}}_{\alpha }}\xi _{{{\alpha }^{-1}}}^{-1}(h)\circledast \varepsilon][{{1}_{\alpha }}\circledast {{\mathscr{A}}_{*}}(\xi _{\alpha }^{*}(F))] \\ &=[{{\varphi }_{\alpha }}{{S}_{{{\alpha }^{-1}}}}(\xi _{{{\alpha }^{-1}}}^{-1}(h))\circledast \varepsilon][{{1}_{\alpha }}\circledast {{\mathscr{A}}_{*}}(\xi _{\alpha }^{*}(F))] \\ \end{align} $

(3.10)

for any $h\in\bar{ H}_{\alpha }$ and $F\in H^{*t, cop}, $ where $\mathscr{A}_{*}$ is the antipode of $H^{*t, cop}$ and $\bar{S}_{\alpha }=\varphi_{\alpha }\circ S_{\alpha ^{-1}}$ is the antipode of $\bar{H};$

$ \bullet $ For any $\alpha \in G, $ the conjugation isomorphism is given by

$ \varphi_{\beta }(h\circledast f)= [\varphi_{\beta }(h)\circledast \varphi_{H^{*t, cop}, \beta }(f)] =[\varphi_{\beta }(h)\circledast \varphi_{\beta^{-1}}^*(f)] $

(3.11)

for any $h\in \bar{H}_{\alpha }$ and $f\in H_{\gamma }^{*t, cop}, $ with $\gamma \in G.$

Proof First, for any $\alpha \in G, $ we will show that $D_{\alpha }(H)$ is an Hom-associative algebra with unit. Then we will show that $\Delta$, defined as above, is multipilcative, i.e., that any $\Delta_{\alpha , \beta }$ is an algebra map. After that, we show that $\varepsilon$ is an algebra map. Finally, we will check axioms for the antipode and the conjugation isomorphisms are compatible with the multiplication.

Hom-associativity Let $\alpha $ be in $G$. The multiplication definition eq.(3.7) is associative if and only if, for any $h, k, l\in (H_{\alpha ^{-1}}, \xi_{\alpha ^{-1}}), f\in (H_{\beta }^{*}, \xi_{\beta }^{*-1}), q\in (H_{\delta}^{*}, \xi_{\delta}^{*-1}), $ and $ p\in (H_{\gamma }^{*}, \xi_{\gamma }^{*-1})$ with $\beta, \delta, \gamma \in G, $

$ ((h\circledast f)(k\circledast q))\xi_{D_{\alpha }(H)}(l\circledast p) =\xi_{D_{\alpha }(H)}(h\circledast f)((k\circledast q)(l\circledast p)). $

(3.12)

By computing the left-hand side of (3.12), we obtain

$ \begin{align} &((h\circledast f)(k\circledast q)){{\xi }_{{{D}_{\alpha }}(H)}}(l\circledast p)= \\ &\xi _{{{\alpha }^{-1}}}^{2}({{(\xi _{{{\alpha }^{-1}}}^{2}({{h}_{(12, {{\alpha }^{-1}})}})k)}_{(12, {{\alpha }^{-1}})}}){{\xi }_{{{\alpha }^{-1}}}}(l)\circledast (f\left\langle {{q}_{11}}, {{\varphi }_{\alpha }}({{h}_{(11, {{\alpha }^{-1}}\delta \alpha )}}) \right\rangle \\ &\left\langle {{q}_{2}}, S_{\delta }^{-1}({{h}_{(2, {{\delta }^{-1}})}}) \right\rangle \xi _{\delta }^{*-2}({{q}_{12}}))\left\langle \xi _{\gamma }^{*-1}({{p}_{11}}), {{\varphi }_{\alpha }}({{(\xi _{{{\alpha }^{-1}}}^{2}({{h}_{(12, {{\alpha }^{-1}})}})k)}_{(11, {{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \\ &\left\langle \xi _{\gamma }^{*-1}({{p}_{2}}), S_{\gamma }^{-1}({{(\xi _{{{\alpha }^{-1}}}^{2}({{h}_{(12, {{\alpha }^{-1}})}})k)}_{(2, {{\gamma }^{-1}})}}) \right\rangle \xi _{\gamma }^{*-2}(\xi _{\gamma }^{*-1}({{p}_{12}})), \\ \end{align} $

by the antimultiplicativity of $S$ and the multiplicativity of $\varphi$

$ \begin{align} &\text{=}\xi _{{{\alpha }^{-1}}}^{5}({{h}_{(1212,{{\alpha }^{-1}})}})(\xi _{{{\alpha }^{-1}}}^{2}({{k}_{(12,{{\alpha }^{-1}})}})l)\circledast \left\langle {{q}_{11}},{{\varphi }_{\alpha }}({{h}_{(11,{{\alpha }^{-1}}\delta \alpha )}}) \right\rangle \left\langle {{q}_{2}},S_{\delta }^{-1}({{h}_{(2,{{\delta }^{-1}})}}) \right\rangle \\ &\left\langle \xi _{\gamma }^{*-1}({{p}_{111}}),{{\varphi }_{\alpha }}({{k}_{(11,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \left\langle \xi _{\gamma }^{*-1}({{p}_{112}}),{{\varphi }_{\alpha }}(\xi _{{{\alpha }^{-1}}\gamma \alpha }^{2}({{h}_{(1211,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \\ &\left\langle \xi _{\gamma }^{*-1}({{p}_{21}}),S_{\gamma }^{-1}(\xi _{{{\gamma }^{-1}}}^{2}({{h}_{(122,{{\gamma }^{-1}})}})) \right\rangle \left\langle \xi _{\gamma }^{*-1}({{p}_{22}}),S_{\gamma }^{-1}({{k}_{(2,{{\gamma }^{-1}})}}) \right\rangle \xi _{\beta }^{*-1}(f)(\xi _{\delta }^{*-2}({{q}_{12}})\xi _{\gamma }^{*-2}({{p}_{12}})) \\ &\text{=}\xi _{{{\alpha }^{-1}}}^{3}({{h}_{(12,{{\alpha }^{-1}})}})(\xi _{{{\alpha }^{-1}}}^{2}({{k}_{(12,{{\alpha }^{-1}})}})l)\circledast \left\langle {{q}_{11}},{{\varphi }_{\alpha }}({{\xi }_{{{\alpha }^{-1}}\delta \alpha }}({{h}_{\left( 111,{{\alpha }^{-1}}\delta \alpha \right)}}) \right\rangle \left\langle {{q}_{2}},S_{\delta }^{-1}\left( {{\xi }_{{{\delta }^{-1}}}}({{h}_{(22,{{\delta }^{-1}})}}) \right) \right\rangle \\ &\left\langle \xi _{\gamma }^{*-1}({{p}_{111}}),{{\varphi }_{\alpha }}({{k}_{(11,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \left\langle \xi _{\gamma }^{*-1}({{p}_{112}}),{{\varphi }_{\alpha }}(\xi _{{{\alpha }^{-1}}\gamma \alpha }^{2}({{h}_{(112,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \\ &\left\langle \xi _{\gamma }^{*-1}({{p}_{21}}),S_{\gamma }^{-1}(\xi _{{{\gamma }^{-1}}}^{2}({{h}_{(21,{{\gamma }^{-1}})}})) \right\rangle \left\langle \xi _{\gamma }^{*-1}({{p}_{22}}),S_{\gamma }^{-1}({{k}_{(2,{{\gamma }^{-1}})}}) \right\rangle \xi _{\beta }^{*-1}(f)(\xi _{\delta }^{*-2}({{q}_{12}})\xi _{\gamma }^{*-2}({{p}_{12}})) \\ \end{align} $

while, by computing the right-hand side, we obtain

$ \begin{align} &{{\xi }_{{{D}_{\alpha }}(H)}}(h\circledast f)((k\circledast q)(l\circledast p)) \\ &=\xi _{{{\alpha }^{-1}}}^{3}({{h}_{(12,{{\alpha }^{-1}})}})(\xi _{{{\alpha }^{-1}}}^{2}({{k}_{(12,{{\alpha }^{-1}})}})l)\circledast \ \left\langle {{p}_{11}},{{\varphi }_{\alpha }}({{k}_{(11,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \ \left\langle {{p}_{2}},S_{\gamma }^{-1}({{k}_{(2,{{\gamma }^{-1}})}})n \right\rangle \\ &\left\langle {{(q\xi _{\gamma }^{*-2}({{p}_{12}}))}_{11}},{{\varphi }_{\alpha }}({{\xi }_{{{\alpha }^{-1}}}}{{(h)}_{(11,{{\alpha }^{-1}}\delta \gamma \alpha )}}))\ \right\rangle \left\langle {{(q\xi _{\gamma }^{*-2}({{p}_{12}}))}_{2}},S_{\gamma \delta }^{-1}({{\xi }_{{{\alpha }^{-1}}}}{{(h)}_{(2,{{\delta }^{-1}}{{\gamma }^{-1}})}})) \right\rangle \\ &\xi _{\beta }^{*-1}(f)(\xi _{\delta }^{*-2}({{q}_{12}})\xi _{\gamma }^{*-4}({{p}_{1212}})) \\ \end{align} $

by the antimultiplicativity of $S$ and the comultiplicativity of $\varphi$

$ \begin{align} &=\xi _{{{\alpha }^{-1}}}^{3}({{h}_{(12,{{\alpha }^{-1}})}})(\xi _{{{\alpha }^{-1}}}^{2}({{k}_{(12,{{\alpha }^{-1}})}})l)\circledast \left\langle {{p}_{11}},{{\varphi }_{\alpha }}({{k}_{(11,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \left\langle {{p}_{2}},S_{\gamma }^{-1}({{k}_{(2,{{\gamma }^{-1}})}}) \right\rangle \\ &\left\langle {{q}_{11}},{{\varphi }_{\alpha }}({{\xi }_{{{\alpha }^{-1}}\delta \alpha }}({{h}_{(111,{{\alpha }^{-1}}\delta \alpha )}})) \right\rangle \left\langle \xi _{\gamma }^{*-2}({{p}_{1211}})),{{\varphi }_{\alpha }}({{\xi }_{{{\alpha }^{-1}}\gamma \alpha }}({{h}_{(112,{{\alpha }^{-1}}\gamma \alpha )}})) \right\rangle \\ &\left\langle {{q}_{2}},S_{\delta }^{-1}({{\xi }_{{{\delta }^{-1}}}}({{h}_{(22,{{\delta }^{-1}})}})) \right\rangle \left\langle \xi _{\gamma }^{*-2}({{p}_{122}})),S_{\gamma }^{-1}({{\xi }_{{{\gamma }^{-1}}}}({{h}_{(21,{{\gamma }^{-1}})}})) \right\rangle \\ &\xi _{\beta }^{*-1}(f)(\xi _{\delta }^{*-2}({{q}_{12}})\xi _{\gamma }^{*-4}({{p}_{1212}})) \\ &=\xi _{{{\alpha }^{-1}}}^{3}({{h}_{(12,{{\alpha }^{-1}})}})(\xi _{{{\alpha }^{-1}}}^{2}({{k}_{(12,{{\alpha }^{-1}})}})l)\circledast \left\langle {{q}_{11}},{{\varphi }_{\alpha }}({{\xi }_{{{\alpha }^{-1}}\delta \alpha }}({{h}_{(111,{{\alpha }^{-1}}\delta \alpha )}})) \right\rangle \left\langle {{q}_{2}},S_{\delta }^{-1}({{\xi }_{{{\delta }^{-1}}}}({{h}_{(22,{{\delta }^{-1}})}})) \right\rangle \\ &\left\langle \xi _{\gamma }^{*-1}({{p}_{111}}),{{\varphi }_{\alpha }}({{k}_{(11,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \left\langle \xi _{\gamma }^{*-1}({{p}_{112}})),{{\varphi }_{\alpha }}({{\xi }_{{{\alpha }^{-1}}\gamma \alpha }}({{h}_{(112,{{\alpha }^{-1}}\gamma \alpha )}})) \right\rangle \\ &\left\langle \xi _{\gamma }^{*-1}({{p}_{21}})),S_{\gamma }^{-1}({{\xi }_{{{\gamma }^{-1}}}}({{h}_{(21,{{\gamma }^{-1}})}})) \right\rangle \left\langle \xi _{\gamma }^{*-1}({{p}_{22}}),S_{\gamma }^{-1}({{k}_{(2,{{\gamma }^{-1}})}}) \right\rangle \xi _{\beta }^{*-1}(f)(\xi _{\delta }^{*-2}({{q}_{12}})\xi _{\gamma }^{*-2}({{p}_{12}})). \\ \end{align} $

Unit Let $\alpha $ be in $G$. For any $h\in H_{\alpha ^{-1}}$ and $f\in H_{\gamma }^{*}$, with $\gamma \in G$, we have

$ \begin{align} &({{1}_{{{\alpha }^{-1}}}}\circledast \varepsilon )(h\circledast f)\text{=}{{\xi }_{{{\alpha }^{-1}}}}(h)\circledast \xi _{\delta }^{*-1}(f) \\ &\text{=}\xi _{{{\alpha }^{-1}}}^{2}({{h}_{(12,{{\alpha }^{-1}})}}){{1}_{{{\alpha }^{-1}}}}\circledast f{{\varepsilon }_{2}}(S_{1}^{-1}({{h}_{(2,1)}})){{\varepsilon }_{11}}({{\varphi }_{\alpha }}({{h}_{(11,1)}}))\xi _{\delta }^{*-2}({{\varepsilon }_{12}}) \\ &=(h\circledast f)({{1}_{{{\alpha }^{-1}}}}\circledast \varepsilon ). \\ \end{align} $

Remark 3.6 Where we use the fact that both $S_{1}$ and $\varphi_{\alpha }$ commute with $\varepsilon$.

Multiplicativity of $\Delta$ Let us prove that $\Delta_{\alpha , \beta }$ is an algebra map for any $\alpha , \beta \in G$. For any $h, k\in H_{\beta^{-1}\alpha ^{-1}}, f\in H_{\gamma }^{*}$ and $g\in H_{\delta }^*$, with $\gamma , \delta\in G, $ we have

$ \Delta_{\alpha , \beta }((h\circledast f)(k\circledast g)) =\Delta_{\alpha , \beta }(h\circledast f)\Delta_{\alpha , \beta }(k\circledast g).\label{comultiplication} $

(3.13)

This is proved by evaluating both terms in the above equation (3.13) against the general term $p\otimes x \otimes q\otimes y$ ($p\in H_{\alpha ^{-1}}^{*}, q\in H_{\beta^{-1}}^{*}, $ and $x, y\in H_{\gamma \delta }$).

Multiplicativity of $\varepsilon$ Let us prove that $\varepsilon$ is an algebra map for any $h, k\in H_{1}, f\in H_{\gamma }^{*}, $ and $g\in H_{\delta}^{*}, $ with $\gamma , \delta\in G$,

$ \left\langle \varepsilon, h\circledast f \right\rangle \left\langle \varepsilon, k\circledast g \right\rangle =\left\langle \varepsilon, h \right\rangle \left\langle f, {{1}_{\gamma }} \right\rangle \left\langle \varepsilon, k \right\rangle \left\langle g, {{1}_{\delta }} \right\rangle $

and

$ \begin{align} &\left\langle \varepsilon ,(h\circledast f)(k\circledast g) \right\rangle =\left\langle \varepsilon ,\xi _{1}^{2}({{h}_{(12,1)}})k\circledast f\left\langle {{g}_{11}},{{h}_{(11,\delta )}}) \right\rangle \left\langle {{g}_{2}},S_{\delta }^{-1}({{h}_{2,{{\delta }^{-1}}}})\xi _{\delta }^{*-2}({{g}_{12}}) \right\rangle \right. \\ &=\left\langle \varepsilon ,\xi _{1}^{2}({{h}_{12}}) \right\rangle \left\langle \varepsilon ,k \right\rangle \left\langle f,{{1}_{\gamma }} \right\rangle \left\langle {{g}_{11}},{{h}_{(11,\delta )}}) \right\rangle \left\langle {{g}_{2}},S_{\delta }^{-1}({{h}_{(2,{{\delta }^{-1}})}}) \right\rangle \left\langle \xi _{\delta }^{*-2}({{g}_{12}}),{{1}_{\delta }} \right\rangle \\ &=\left\langle \varepsilon ,k \right\rangle \left\langle f,{{1}_{\gamma }} \right\rangle \left\langle g,S_{\delta }^{-1}({{h}_{(2,{{\delta }^{-1}})}}){{h}_{(1,\delta )}}) \right\rangle =\left\langle \varepsilon ,h \right\rangle \left\langle f,{{1}_{\gamma }} \right\rangle \left\langle \varepsilon ,k \right\rangle \left\langle g,{{1}_{\delta }} \right\rangle . \\ \end{align} $

This proves that $\varepsilon$ is multiplicative. Moreover, since $\varepsilon$ is obviously unitary, it is an algebra homomorphism.

Antipode Let $h\in H_{1}$ and let $f\in H_{\gamma }^{*}, $ with $\gamma \in G$,

$ \begin{align} &{{(h\circledast f)}_{(1,\alpha )}}{{S}_{{{\alpha }^{-1}}}}({{(h\circledast f)}_{(2,{{\alpha }^{-1}})}}) \\ &\text{=}({{\varphi }_{{{\alpha }^{-1}}}}({{h}_{(1,{{\alpha }^{-1}})}})\circledast {{f}_{1}})[(({{\varphi }_{{{\alpha }^{-1}}}}{}^\circ {{S}_{\alpha }})\xi _{\alpha }^{-1}({{h}_{(2,\alpha )}})\circledast \varepsilon )({{1}_{{{\alpha }^{-1}}}}\circledast {{\mathcal{A}}_{*}}(\xi _{\gamma }^{*}({{f}_{2}})))] \\ &\text{=}({{\varphi }_{{{\alpha }^{-1}}}}(\xi _{{{\alpha }^{-1}}}^{-1}({{h}_{(1,{{\alpha }^{-1}})}}){{S}_{\alpha }}\xi _{\alpha }^{-1}({{h}_{(2,\alpha )}}))\circledast \xi _{\gamma }^{*}({{f}_{1}})\varepsilon )({{1}_{{{\alpha }^{-1}}}}\circledast {{\mathcal{A}}_{*}}({{f}_{2}})) \\ &\text{=}\left\langle \varepsilon ,h\circledast f \right\rangle {{1}_{{{\alpha }^{-1}}}}\circledast \varepsilon \\ \end{align} $

and

$ \begin{align} &{{S}_{{{\alpha }^{-1}}}}({{(h\circledast f)}_{(1,{{\alpha }^{-1}})}}){{(h\circledast f)}_{(2,\alpha )}} \\ &\text{=}[({{S}_{\alpha }}\xi _{\alpha }^{-1}({{h}_{(1,\alpha )}})\circledast \varepsilon )({{1}_{{{\alpha }^{-1}}}}\circledast {{\mathcal{A}}_{*}}(\xi _{\gamma }^{*}({{f}_{1}})))]({{h}_{(2,{{\alpha }^{-1}})}}\circledast {{f}_{2}}) \\ &\text{=}({{S}_{\alpha }}({{h}_{(1,\alpha )}})\circledast \varepsilon )({{h}_{(2,{{\alpha }^{-1}})}}\circledast {{\mathcal{A}}_{*}}(\xi _{\gamma }^{*}({{f}_{1}}))\left\langle \xi _{\gamma }^{*}{{({{f}_{2}})}_{11}},{{1}_{\gamma }} \right\rangle \left\langle \xi _{\gamma }^{*}{{({{f}_{2}})}_{2}},{{1}_{\gamma }} \right\rangle \xi _{\gamma }^{*-2}(\xi _{\gamma }^{*}{{({{f}_{2}})}_{12}}) \\ &\text{=}\left\langle f,{{1}_{\gamma }} \right\rangle {{S}_{\alpha }}({{h}_{(1,\alpha )}}){{h}_{(2,{{\alpha }^{-1}})}}\circledast \varepsilon \\ &\text{=}\left\langle \varepsilon ,h\circledast f \right\rangle {{1}_{{{\alpha }^{-1}}}}\circledast \varepsilon . \\ \end{align} $

Conjugation Let us check that $\varphi_{\beta }$ is an algebra isomorphism for any $\alpha , \beta \in G.$ For all $h, k\in H_{\alpha }^{-1}, f\in H_{\gamma }^{*}, $ and $g\in H_{\delta}^{*}, $ with $\gamma , \delta\in G, $

$ \begin{align} &({{\varphi }_{\beta }}(h\circledast f){{\varphi }_{\beta }}(k\circledast g))(p\otimes x) \\ &=(\xi _{\beta {{\alpha }^{-1}}{{\beta }^{-1}}}^{2}({{\varphi }_{\beta }}{{(h)}_{12}}){{\varphi }_{\beta }}(k)\circledast \varphi _{{{\beta }^{-1}}}^{*}(f)\left\langle \varphi _{{{\beta }^{-1}}}^{*}{{(g)}_{11}},{{\varphi }_{\alpha }}({{\varphi }_{\beta }}{{(h)}_{(11,{{\alpha }^{-1}}{{\beta }^{-1}}\delta \beta \alpha )}}) \right\rangle \\ &\left\langle \varphi _{{{\beta }^{-1}}}^{*}{{(g)}_{2}},S_{\beta \delta {{\beta }^{-1}}}^{-1}({{\varphi }_{\beta }}{{(h)}_{(2,\beta {{\delta }^{-1}}{{\beta }^{-1}})}})) \right\rangle \xi _{{{\beta }^{-1}}\delta \beta }^{*-2}(\varphi _{{{\beta }^{-1}}}^{*}{{(g)}_{12}}))(p\otimes x) \\ &={{\varphi }_{\beta }}(\xi _{{{\alpha }^{-1}}}^{2}({{h}_{12}})k)p\circledast \left\langle {{g}_{11}},{{\varphi }_{\alpha }}({{h}_{(11,{{\alpha }^{-1}}\delta \alpha )}}) \right\rangle \left\langle {{g}_{2}},S_{\delta }^{-1}({{h}_{(2,{{\delta }^{-1}})}}) \right\rangle \\ &\varphi _{{{\beta }^{-1}}}^{*}(f\xi _{\delta }^{*-2}({{g}_{12}}))(x) \\ &={{\varphi }_{\beta }}((h\circledast f)(k\circledast g))(p\otimes x) \\ \end{align} $

for all $x\in H_{\beta\gamma \delta\beta ^{-1}}, p\in H_{\beta\alpha ^{-1}\beta^{-1}}^{*}$.

For any $\alpha \in G, $ we set $n_{\alpha }=\dim H_{\alpha }.$ Let $(\kappa_{(\alpha , i)})_{i=1, \cdots, n_{\alpha }}$ and $(\kappa^{(\alpha , i)})_{i=1, \cdots, n_{\alpha }}$ be dual bases in $H_{\alpha }$ and $H_{\alpha }^*.$ Then we have the following proposition.

Definition 3.7 A quasitriangular Hom-Hopf $T$-coalgebra is a Hom-Hopf $T$-coalgebra endowed with a family $R=\{R_{\alpha , \beta}=\kappa_{\alpha , i}\otimes \kappa_{\beta, i} \in H_{\alpha }\otimes H_{\beta}\}_{\alpha , \beta\in G}, $ called a universal $R$-matrix, such that $R_{\alpha , \beta}$ is invertible for any $\alpha , \beta\in G$ and the following conditions are satisfied:

$ R_{\alpha , \beta}\Delta_{\alpha , \beta}(h) =(\tau\circ(\varphi_{\alpha ^{-1}}\otimes id_{\alpha }) \circ \Delta_{\alpha \beta\alpha ^{-1}, \alpha })(h)R_{\alpha , \beta}, $

(3.14)

$ (\xi_{\alpha }\otimes \xi_{\beta })R_{\alpha , \beta } =R_{\alpha , \beta } $

(3.15)

for all $h\in H_{\alpha \beta}$ and $\alpha , \beta\in G, $

$ \kappa_{\alpha , i}\otimes \kappa_{(1, \beta ), i}\otimes \kappa_{(2, \gamma ), i} = \kappa_{\alpha , i}\kappa_{\alpha , j}\otimes \kappa_{\beta, j}\otimes \kappa_{\gamma , i}, $

(3.16)

$ \kappa_{(1, \alpha ), i}\otimes \kappa_{(2, \beta ), i}\otimes \kappa_{\gamma , i} = \varphi_{\beta }(\kappa_{\beta^{-1}\alpha \beta , i})\otimes \kappa_{\beta, j}\otimes \kappa_{\gamma , i}\kappa_{\gamma , j}, $

(3.17)

$ (\varphi_{\beta}\otimes \varphi_{\beta})(R_{\alpha , \gamma }) =R_{\beta\alpha \beta^{-1}, \beta\gamma \beta^{-1}} $

(3.18)

for all $\alpha , \beta, \gamma \in G.$

Remark 3.8 (1) We introduce the notation $\tilde{\kappa}_{\alpha , i}\otimes \tilde{\kappa}_{\beta, i} =\tilde{R}_{\alpha , \beta}=(R^{-1})_{\alpha , \beta}.$ (2) $R_{1, 1}$ is an $R$-matrix for the Hom-Hopf algebra $H_{1}$ (see [8]).

Proposition 3.9 The Drinfeld double $D(H)=\{D_{\alpha }(H)\}_{\alpha \in G}$ has a quasitriangular structure given by

$ R_{\alpha , \beta }=(\kappa_{(\alpha ^{-1}, i)}\circledast \varepsilon) \otimes (1_{\beta^{-1}}\circledast\kappa^{(\alpha ^{-1}, i)}) \in D_{\alpha }(H)\otimes D_{\beta }(H) $

(3.19)

and

$ \bar{R}_{\alpha , \beta }=(S_{\alpha }(\kappa_{(\alpha , i)})\circledast \varepsilon) \otimes (1_{\beta^{-1}}\circledast\kappa^{(\alpha , i)}) \in D_{\alpha }(H)\otimes D_{\beta }(H). $

(3.20)

Proof Relation (3.14):

$ R_{\alpha , \beta}\Delta_{\alpha , \beta}(h) =(\tau\circ(\varphi_{\alpha ^{-1}}\otimes id_{\alpha }) \circ \Delta_{\alpha \beta\alpha ^{-1}, \alpha })(h)R_{\alpha , \beta}, $

Let $\alpha , \beta , \gamma \in G.$ Given $h\in H_{\beta^{-1}\alpha ^{-1}}$ and $f\in H_{\gamma }^{*}, $ we have

$ \begin{align} &{{R}_{\alpha ,\beta }}{{\Delta }_{\alpha ,\beta }}(h\circledast f) \\ &=(({{\kappa }_{({{\alpha }^{-1}},i)}}\circledast \varepsilon )\otimes ({{1}_{{{\beta }^{-1}}}}\otimes {{\kappa }^{({{\alpha }^{-1}},i)}}))(({{\varphi }_{\beta }}({{h}_{(1,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}})\circledast {{f}_{1}})\otimes ({{h}_{(2,{{\beta }^{-1}})}}\circledast {{f}_{2}})) \\ &=(\xi _{{{\alpha }^{-1}}}^{2}({{({{\kappa }_{({{\alpha }^{-1}},i)}})}_{12}}){{\varphi }_{\beta }}({{h}_{(1,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}})\circledast \varepsilon \left\langle {{f}_{111}},{{\varphi }_{\alpha }}({{({{\kappa }_{({{\alpha }^{-1}},i)}})}_{(11,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \\ &\left\langle {{f}_{12}},S_{\gamma }^{-1}({{({{\kappa }_{({{\alpha }^{-1}},i)}})}_{(2,{{\gamma }^{-1}})}}) \right\rangle \xi _{\gamma }^{*-2}({{f}_{112}}))\otimes (\xi _{{{\beta }^{-1}}}^{2}({{1}_{{{\beta }^{-1}}}}){{h}_{(2,{{\beta }^{-1}})}}\circledast {{\kappa }^{({{\alpha }^{-1}},i)}}) \\ &\left\langle {{f}_{211}},{{1}_{\gamma }} \right\rangle \left\langle _{22},{{1}_{\gamma }} \right\rangle f\xi _{\gamma }^{*-2}({{f}_{212}}) \\ &=(\xi _{{{\alpha }^{-1}}}^{2}({{({{\kappa }_{({{\alpha }^{-1}},i)}})}_{12}}){{\varphi }_{\beta }}({{h}_{(1,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}})\circledast \left\langle {{f}_{111}},{{\varphi }_{\alpha }}({{({{\kappa }_{({{\alpha }^{-1}},i)}})}_{(11,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \\ &\left\langle {{f}_{12}},S_{\gamma }^{-1}({{({{\kappa }_{({{\alpha }^{-1}},i)}})}_{(2,{{\gamma }^{-1}})}}) \right\rangle \xi _{\gamma }^{*-3}({{f}_{112}}))\otimes ({{\xi }_{{{\beta }^{-1}}}}({{h}_{2,{{\beta }^{-1}}}})\circledast {{\kappa }^{({{\alpha }^{-1}},i)}}{{f}_{2}}) \\ \end{align} $

(19)

and

$ \begin{align} &((\tau {}^\circ ({{\varphi }_{{{\alpha }^{-1}}}}\otimes i{{d}_{{{D}_{\alpha }}(H)}}){}^\circ {{\Delta }_{\alpha \beta {{\alpha }^{-1}},\alpha }})(h\circledast f)){{R}_{\alpha ,\beta }} \\ &=({{h}_{(2,{{\alpha }^{-1}})}}\circledast {{f}_{2}})({{\kappa }_{({{\alpha }^{-1}},i)}}\circledast \varepsilon )\otimes ({{h}_{(1,{{\beta }^{-1}})}}\circledast \varphi _{\alpha }^{*}({{f}_{1}}))({{1}_{{{\beta }^{-1}}}}\otimes {{\kappa }^{({{\alpha }^{-1}},i)}}) \\ &=(\xi _{{{\alpha }^{-1}}}^{2}({{h}_{(212,{{\alpha }^{-1}})}}){{\kappa }_{{{\alpha }^{-1}},i}}\circledast {{f}_{2}}\left\langle \varepsilon ,S_{1}^{-1}({{h}_{(22,1)}}) \right\rangle \left\langle \varepsilon ,{{\varphi }_{\alpha }}({{h}_{(211,1)}}) \right\rangle \xi _{1}^{*-2}(\varepsilon )) \\ &\otimes (\xi _{{{\beta }^{-1}}}^{2}({{h}_{(112,{{\beta }^{-1}})}}){{1}_{{{\beta }^{-1}}}}\circledast \varphi _{\alpha }^{*}({{f}_{1}})\left\langle \kappa _{11}^{({{\alpha }^{-1}},i)},{{\varphi }_{\beta }}({{h}_{(111,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}}) \right\rangle \\ &\left\langle \kappa _{2}^{({{\alpha }^{-1}},i)},S_{{{\alpha }^{-1}}}^{-1}({{h}_{(12,\alpha )}}) \right\rangle \xi _{{{\alpha }^{-1}}}^{*-2}(\kappa _{12}^{({{\alpha }^{-1}},i)})) \\ &=({{h}_{(2,{{\alpha }^{-1}})}}{{\kappa }_{({{\alpha }^{-1}},i)}}\circledast \xi _{\gamma }^{*-1}({{f}_{2}}))\otimes (\xi _{{{\beta }^{-1}}}^{3}({{h}_{(112,{{\beta }^{-1}})}})\circledast \left\langle \kappa _{11}^{({{\alpha }^{-1}},i)},{{\varphi }_{\beta }}({{h}_{(111,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}}) \right\rangle \\ &\left\langle \kappa _{2}^{({{\alpha }^{-1}},i)},S_{{{\alpha }^{-1}}}^{-1}({{h}_{(12,\alpha )}}) \right\rangle \varphi _{\alpha }^{*}({{f}_{1}})\xi _{{{\alpha }^{-1}}}^{*-2}(\kappa _{12}^{({{\alpha }^{-1}},i)})). \\ \end{align} $

Relation (3.14) is proved by observing that evaluating the two expressions above against the tensor $i{{d}_{{{\alpha }^{-1}}}}\otimes id_{\gamma }^{*}\otimes i{{d}_{{{\beta }^{-1}}}}\otimes \left\langle \cdot ,x \right\rangle $ (for $x\in H_{\alpha ^{-1}\gamma }$) we get the same result

$ \begin{align} &(\xi _{{{\alpha }^{-1}}}^{2}({{({{\kappa }_{({{\alpha }^{-1}},i)}})}_{12}}){{\varphi }_{\beta }}({{h}_{(1,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}})\circledast \left\langle {{f}_{111}},{{\varphi }_{\alpha }}({{({{\kappa }_{({{\alpha }^{-1}},i)}})}_{(11,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \\ &\left\langle {{f}_{12}},S_{\gamma }^{-1}({{({{\kappa }_{({{\alpha }^{-1}},i)}})}_{(2,{{\gamma }^{-1}})}}) \right\rangle \xi _{\gamma }^{*-3}({{f}_{112}}))\otimes ({{\xi }_{{{\beta }^{-1}}}}({{h}_{(2,{{\beta }^{-1}})}})\circledast \left\langle {{\kappa }^{({{\alpha }^{-1}},i)}}{{f}_{2}},x \right\rangle ) \\ &=({{x}_{(2,{{\alpha }^{-1}})}}{{\varphi }_{\beta }}({{h}_{(1,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}})\circledast \left\langle {{f}_{1}},{{\varphi }_{\alpha }}({{x}_{(1,{{\alpha }^{-1}}\gamma \alpha )}}) \right\rangle \xi _{\gamma }^{*-1}({{f}_{2}}))\otimes {{\xi }_{{{\beta }^{-1}}}}({{h}_{(2,{{\beta }^{-1}})}}) \\ &=({{h}_{(2,{{\alpha }^{-1}})}}{{\kappa }_{({{\alpha }^{-1}},i)}}\circledast \xi _{\gamma }^{*-1}({{f}_{2}}))\otimes (\xi _{{{\beta }^{-1}}}^{3}({{h}_{(112,{{\beta }^{-1}})}})\circledast \left\langle \kappa _{11}^{({{\alpha }^{-1}},i)},{{\varphi }_{\beta }}({{h}_{(111,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}}) \right\rangle \\ &\left\langle \kappa _{2}^{({{\alpha }^{-1}},i)},S_{{{\alpha }^{-1}}}^{-1}({{h}_{(12,\alpha )}})\varphi _{\alpha }^{*}({{f}_{1}})\xi _{{{\alpha }^{-1}}}^{*-2}(\kappa _{12}^{({{\alpha }^{-1}},i)}),x \right\rangle ), \\ \end{align} $

where we used

$ \sum\limits_{i}{\left\langle f,\text{ }{{\kappa }_{({{\alpha }^{-1}},i)}}{{\kappa }^{({{\alpha }^{-1}},i)}} \right\rangle }=f\quad \text{ and }\quad \sum\limits_{i}{\left\langle {{\kappa }^{({{\alpha }^{-1}},i)}},h \right\rangle }{{\kappa }_{({{\alpha }^{-1}},i)}}=h $

for all $f\in H_{\alpha ^{-1}}^{*}, h\in H_{\alpha ^{-1}}$ and $\alpha \in G.$

Then we check Relation (3.16) and (3.17). The identities

$ \begin{align} &({{\kappa }_{({{\alpha }^{-1}},i)}}\circledast \varepsilon )\otimes ({{1}_{{{\beta }^{-1}}}}\circledast \kappa _{1}^{({{\alpha }^{-1}},i)})\otimes ({{1}_{{{\gamma }^{-1}}}}\circledast \kappa _{2}^{({{\alpha }^{-1}},i)}) \\ &=({{\kappa }_{({{\alpha }^{-1}},i)}}{{\kappa }_{({{\alpha }^{-1}},j)}}\circledast \varepsilon )\otimes ({{1}_{{{\beta }^{-1}}}}\circledast {{\kappa }^{({{\alpha }^{-1}},j)}})\otimes ({{1}_{{{\gamma }^{-1}}}}\circledast {{\kappa }^{({{\alpha }^{-1}},i)}}) \\ \end{align} $

and

$ \begin{align} &({{\varphi }_{\beta }}({{({{\kappa }_{({{(\alpha \beta )}^{-1}},i)}})}_{(1,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}})\circledast \varepsilon )\otimes ({{({{\kappa }_{({{(\alpha \beta )}^{-1}},i)}})}_{(2,{{\beta }^{-1}})}}\circledast \varepsilon )\otimes ({{1}_{{{\gamma }^{-1}}}}\circledast {{\kappa }^{({{(\alpha \beta )}^{-1}},i)}}) \\ &=({{\varphi }_{\beta }}({{\kappa }_{({{\beta }^{-1}}{{\alpha }^{-1}}\beta ,i)}})\circledast \varepsilon )\otimes ({{\kappa }_{({{\beta }^{-1}},j)}}\circledast \varepsilon )\otimes ({{1}_{{{\gamma }^{-1}}}}\circledast {{\kappa }^{({{\beta }^{-1}}{{\alpha }^{-1}}\beta ,i)}}{{\kappa }^{({{\beta }^{-1}},j)}}) \\ \end{align} $

can be written as (identifying $\bar{H_{\alpha }}\otimes \varepsilon$ with $\bar{H_{\alpha }}$ and $1_{\beta^{-1}}\otimes H^*$ with $H^*$)

$ \kappa_{(\alpha ^{-1}, i)}\otimes \kappa_{1}^{(\alpha ^{-1}, i)}\otimes \kappa_{2}^{(\alpha ^{-1}, i)} = \kappa_{(\alpha ^{-1}, i)}\kappa_{(\alpha ^{-1}, j)}\otimes \kappa ^{(\alpha ^{-1}, j)}\otimes \kappa ^{(\alpha ^{-1}, i)}\\ $

(3.21)

$ \begin{align} &{{({{\kappa }_{({{(\alpha \beta )}^{-1}},i)}})}_{(1,{{\beta }^{-1}}{{\alpha }^{-1}}\beta )}}\otimes {{({{\kappa }_{({{(\alpha \beta )}^{-1}},i)}})}_{(2,{{\beta }^{-1}})}}\otimes {{\kappa }^{({{(\alpha \beta )}^{-1}},i)}} \\ &={{\kappa }_{({{\beta }^{-1}}{{\alpha }^{-1}}\beta ,i)}}\otimes {{\kappa }_{({{\beta }^{-1}},j)}}\otimes {{\kappa }^{({{\beta }^{-1}}{{\alpha }^{-1}}\beta ,i)}}{{\kappa }^{({{\beta }^{-1}},j)}}. \\ \end{align} $

(3.22)

The above equalities can be verified by evaluating both sides on element $f\in H_{\alpha ^{-1}}^{*}$ in the first factor (respectively, on $h\in \overline{H}_{\alpha \beta }$ in the third factor) (see Zunino [9], Theorem 11).

Finally, let us check that

$ \begin{align} &({{\xi }_{{{D}_{\alpha }}(H)}}\otimes {{\xi }_{{{D}_{\beta }}(H)}}){{R}_{\alpha ,\beta }}={{R}_{\alpha ,\beta }}, \\ &({{\xi }_{{{D}_{\alpha }}(H)}}\otimes {{\xi }_{{{D}_{\beta }}(H)}}){{R}_{\alpha ,\beta }}={{\xi }_{{{D}_{\alpha }}(H)}}({{\kappa }_{({{\alpha }^{-1}},i)}}\circledast \varepsilon )\otimes {{\xi }_{{{D}_{\beta }}(H)}}({{1}_{{{\beta }^{-1}}}}\circledast {{\kappa }^{({{\alpha }^{-1}},i)}}) \\ &\ \ \ \ \ \ \ \ \ =({{\xi }_{{{\alpha }^{-1}}}}({{\kappa }_{({{\alpha }^{-1}},i)}})\circledast \varepsilon )\otimes ({{1}_{{{\beta }^{-1}}}}\circledast \xi _{{{\alpha }^{-1}}}^{*-1}({{\kappa }^{({{\alpha }^{-1}},i)}})). \\ \end{align} $

Now, $\xi_{\alpha ^{-1}}$ is a linear isomorphism, so $(\xi_{\alpha ^{-1}}(\kappa_{(\alpha ^{-1}, i)}))_{i=1, \cdots, n_{\alpha }}$ is a basis of $H_{\alpha ^{-1}}, $ and

$ (\xi_{\alpha ^{-1}}^{*-1}(\kappa_{(\alpha ^{-1}, i)}))_{i=1, \cdots, n_{\alpha }} $

is its dual basis. So $(\xi_{D_{\alpha }(H)}\otimes \xi_{D_{\beta }(H)})R_{\alpha , \beta } =R_{\alpha , \beta }.$

$ \begin{align} &({{\xi }_{{{\alpha }^{-1}}}}({{\kappa }_{({{\alpha }^{-1}},i)}})\circledast \varepsilon )\otimes ({{1}_{{{\beta }^{-1}}}}\circledast \left\langle \xi _{{{\alpha }^{-1}}}^{*-1}({{\kappa }^{({{\alpha }^{-1}},i)}}),x \right\rangle ) \\ &=({{\xi }_{{{\alpha }^{-1}}}}({{\kappa }_{({{\alpha }^{-1}},i)}}\left\langle {{\kappa }^{({{\alpha }^{-1}},i)}},\xi _{{{\alpha }^{-1}}}^{-1}(x) \right\rangle )\circledast \varepsilon )\otimes {{1}_{{{\beta }^{-1}}}} \\ &=(x\circledast \varepsilon )\otimes {{1}_{{{\beta }^{-1}}}} \\ &=({{\kappa }_{({{\alpha }^{-1}},i)}}\circledast \varepsilon )\otimes ({{1}_{{{\beta }^{-1}}}}\circledast \left\langle {{\kappa }^{({{\alpha }^{-1}},i)}},x \right\rangle ) \\ \end{align} $

for $x\in H_{\alpha ^{-1}}.$

This completes the proof.