A Yetter-Drinfel'd module over a Hopf algebra, firstly introduced by Yetter (crossed bimodule in [1]), is a module and a comodule satisfying a certain compatibility condition. The main feature is that Yetter-Drinfel'd modules form a pre-braided monoidal category. Under favourable conditions (e.g. if the antipode of the Hopf algebra is bijective), the category is even braided (or quasisymmetric). Via a (pre-) braiding structure, the notion of Yetter-Drinfel'd module plays a part in the relations between quantum groups and knot theory.

When a Hopf algebra is finite-dimensional, the generalized (anti) Yetter-Drinfel'd module category was studied in [2]. The authors showed that $ {}_{H}\mathcal{YD}^{H}(\alpha, \beta) \cong {}_{H^* \bowtie H(\alpha, \beta)}\mathcal{M}, $ where $ H^* \bowtie H(\alpha, \beta) $ is the diagonal crossed product algebra. Then one main question naturally arises: Does this isomorphism still hold for some infinite-dimensional Hopf algebra?

For this question, we first recall from our paper [3] the diagonal crossed product of an infinite-dimensional coFrobenius Hopf algebra, then we consider the representation category of the diagonal crossed product, and show that for a coFrobenius Hopf algebra $ H $ with its dual multiplier Hopf algebra $ \widehat{H} $, the unital $ \widehat{H} \bowtie H(\alpha, \beta) $-module category is isomorphic to $ (\alpha, \beta) $-Yetter-Drinfeld module category introduced in [2, 4], i.e., $ {}_{H}\mathcal{YD}^{H}(\alpha, \beta) \cong {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M}. $ Moreover, as braided $ T $-categories the representation category $ Rep(\bigoplus_{(\alpha, \beta)\in G} \widehat{H} \bowtie H(\alpha, \beta)) $ is isomorphic to $ \mathcal{YD}(H) $ introduced in [2].

The paper is organized in the following way. In section 2, we recall some notions which will be used in the following, such as multiplier Hopf algebras and $ (\alpha, \beta) $-quantum double of an infinite dimensional coFrobenius Hopf algebra.

In section 3, we show that for a coFrobenius Hopf algebra $ H $, the unital $ \widehat{H} \bowtie H(\alpha, \beta) $-module category $ {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M} $ is isomorphic to $ {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $. And as braided $ T $-categories the representation theory $ Rep(\mathcal{A}) $ is isomorphic to $ \mathcal{YD}(H) $ introduced in [2], generalizing the classical result in [2, 5].

We begin this section with a short introduction to multiplier Hopf algebras.

Throughout this paper, all spaces we considered are over a fixed field $ K $ (such as the field $ \mathbb{C} $ of complex numbers). Algebras may or may not have units, but always should be non-degenerate, i.e., the multiplication maps (viewed as bilinear forms) are non-degenerate. Recalling from the appendix in [6], the multiplier algebra $ M(A) $ of an algebra $ A $ is defined as the largest algebra with unit in which $ A $ is a dense ideal.

Now, we recall the definition of a multiplier Hopf algebra (see [6] for details). A comultiplication on an algebra $ A $ is a homomorphism $ \Delta: A \longrightarrow M(A \otimes A) $ such that $ \Delta(a)(1 \otimes b) $ and $ (a \otimes 1)\Delta(b) $ belong to $ A\otimes A $ for all $ a, b \in A $. We require $ \Delta $ to be coassociative in the sense that

$ \begin{eqnarray*} (a\otimes 1\otimes 1)(\Delta \otimes \iota)(\Delta(b)(1\otimes c)) = (\iota \otimes \Delta)((a \otimes 1)\Delta(b))(1\otimes 1\otimes c) \end{eqnarray*} $

for all $ a, b, c \in A $ (where $ \iota $ denotes the identity map).

A pair $ (A, \Delta) $ of an algebra $ A $ with non-degenerate product and a comultiplication $ \Delta $ on $ A $ is called a multiplier Hopf algebra, if the maps $ T_{1}, T_{2}: A\otimes A \longrightarrow M(A\otimes A) $ defined by

$ \begin{eqnarray} T_{1}(a\otimes b) = \Delta(a)(1 \otimes b), \qquad T_{2}(a\otimes b) = (a \otimes 1)\Delta(b) \end{eqnarray} $

(2.1)

have range in $ A\otimes A $ and are bijective.

A multiplier Hopf algebra $ (A, \Delta) $ is called regular if $ (A, \Delta^{cop}) $ is also a multiplier Hopf algebra, where $ \Delta^{cop} $ denotes the co-opposite comultiplication defined as $ \Delta^{cop} = \tau \circ \Delta $ with $ \tau $ the usual flip map from $ A\otimes A $ to itself (and extended to $ M(A\otimes A) $). In this case, $ \Delta(a)(b \otimes 1) \mbox{ and } (1 \otimes a)\Delta(b) \in A \otimes A $ for all $ a, b\in A $.

Multiplier Hopf algebra $ (A, \Delta) $ is regular if and only if the antipode $ S $ is bijective from $ A $ to $ A $ (see [7], Proposition 2.9). In this situation, the comultiplication is also determined by the bijective maps $ T_{3}, T_{4}: A\otimes A \longrightarrow A\otimes A $ defined as follows

$ \begin{eqnarray} && T_{3}(a \otimes b) = \Delta(a)(b \otimes 1), \qquad T_{4}(a\otimes b) = (1 \otimes a)\Delta(b) \end{eqnarray} $

(2.2)

for all $ a, b\in A $.

In this paper, all the multiplier hopf algebras we considered are regular. We will use the adapted Sweedler notation for regular multiplier Hopf algebras (see [8]). We will e.g., write $ \sum a_{(1)} \otimes a_{(2)}b $ for $ \Delta(a)(1 \otimes b) $ and $ \sum ab_{(1)} \otimes b_{(2)} $ for $ (a \otimes 1)\Delta(b) $, sometimes we omit the $ \sum $.

Define two linear operators $ \mathcal{T} $ and $ \mathcal{T'} $ acting on $ A \otimes A $ introduced by the formulae

$ \begin{eqnarray*} && \mathcal{T}(a \otimes b) = b_{(2)} \otimes aS(b_{(1)})b_{(3)}, \\ && \mathcal{T'}(a \otimes b) = b_{(1)} \otimes S(b_{(2)})ab_{(3)} \end{eqnarray*} $

for any $ a, b\in A $.

These two operators above are well-defined, since

$ \begin{eqnarray*} && \mathcal{T}(a \otimes b) = T_{4}(S\otimes \iota)T_{3}(\iota\otimes S^{-1})\tau (a\otimes b), \\ && \mathcal{T'}(a \otimes b) = (\iota \otimes S) T_{4} \tau (\iota\otimes S^{-1}) T_{4} (a\otimes b). \end{eqnarray*} $

They are obviously invertible, and the inverses can be written as follows

$ \begin{eqnarray*} && \mathcal{T}^{-1}(a \otimes b) = b S^{-1}(a_{(3)}) a_{(1)} \otimes a_{(2)}, \\ && \mathcal{T'}^{-1}(a \otimes b) = a_{(3)} b S^{-1}(a_{(2)}) \otimes a_{(1)}. \end{eqnarray*} $

For any $ a, b \in A $, $ \mathcal{T} \circ T_{2} = T_{4} $. If $ A $ is commutative, then $ \mathcal{T'} = \tau $, and if $ A $ is cocommutative, then $ \mathcal{T} = \tau $.

Proposition 2.1  Operators $ \mathcal{T} $ and $ \mathcal{T'} $ satisfy the braided equation

$ \begin{eqnarray*} && (\mathcal{T} \otimes \iota)(\iota \otimes \mathcal{T})(\mathcal{T} \otimes \iota) = (\iota \otimes \mathcal{T})(\mathcal{T} \otimes \iota)(\iota \otimes \mathcal{T}), \\ && (\mathcal{T'} \otimes \iota)(\iota \otimes \mathcal{T'})(\mathcal{T'} \otimes \iota) = (\iota \otimes \mathcal{T'})(\mathcal{T'} \otimes \iota)(\iota \otimes \mathcal{T'}). \end{eqnarray*} $

Let $ H $ be a Hopf algebra, and $ \alpha, \beta \in Aut_{Hopf}(H) $. Denote $ G = Aut_{Hopf}(H)\times Aut_{Hopf}(H) $, a group with multiplication $ (\alpha, \beta) \ast (\gamma, \delta) = (\alpha\gamma, \delta\gamma^{-1}\beta\gamma) $. The unit is $ (\iota, \iota) $ and $ (\alpha, \beta)^{-1} = (\alpha^{-1}, \alpha\beta^{-1}\alpha^{-1}) $.

Let $ H $ be a coFrobenius Hopf algebra with its dual multiplier Hopf algebra $ \widehat{H} $. Then $ \mathcal{A} = \bigoplus_{(\alpha, \beta)\in G} \mathcal{A}_{(\alpha, \beta)} = \bigoplus_{(\alpha, \beta)\in G} \widehat{H} \bowtie H(\alpha, \beta) $ is a $ G $-cograded multiplier Hopf algebra with the following strucrures:

● For any $ (\alpha, \beta)\in G $, $ \mathcal{A}_{(\alpha, \beta)} $ has the multiplication given by

$ \begin{eqnarray*} (p\bowtie h)(q\bowtie l) = p \big( \alpha(h_{(1)}) \blacktriangleright q \blacktriangleleft S^{-1}\beta(h_{(3)}) \big) \bowtie h_{(2)}l \end{eqnarray*} $

for $ p, q\in \widehat{H} $ and $ h, l\in H $.

● The comultiplication on $ \mathcal{A} $ is given by:

$ \begin{eqnarray*} && \Delta_{(\alpha, \beta), (\gamma, \delta)}: \mathcal{A}_{(\alpha, \beta)\ast (\gamma, \delta)} \longrightarrow M(\mathcal{A}_{(\alpha, \beta)} \otimes \mathcal{A}_{(\gamma, \delta)}), \\ && \Delta_{(\alpha, \beta), (\gamma, \delta)} (p\bowtie h) = \Delta^{cop}(p)(\gamma\otimes \gamma^{-1}\beta\gamma)\Delta(h). \end{eqnarray*} $

● The counit $ \varepsilon_{\mathcal{A}} $ on $ \mathcal{A}_{(\iota, \iota)} = D(H) $ is the counit on the Drinfel'd double of $ H $.

● For any $ (\alpha, \beta)\in G $, the antipode is given by

$ \begin{eqnarray*} && S: \mathcal{A}_{(\alpha, \beta)} \longrightarrow \mathcal{A}_{(\alpha, \beta)^{-1}}, \\ && S_{(\alpha, \beta)}(p\bowtie h) = T(\alpha\beta S(h) \otimes S^{-1}(p)) \mbox{ in } \mathcal{A}_{(\alpha, \beta)^{-1}} = \mathcal{A}_{(\alpha^{-1}, \alpha\beta^{-1}\alpha^{-1})}. \end{eqnarray*} $

● A crossing action $ \xi: G \longrightarrow Aut(\mathcal{A}) $ is given by

$ \begin{eqnarray*} && \xi_{(\alpha, \beta)}^{(\gamma, \delta)}: \mathcal{A}_{(\gamma, \delta)} \longrightarrow \mathcal{A}_{(\alpha, \beta)\ast(\gamma, \delta)\ast(\alpha, \beta)^{-1}} = \mathcal{A}_{(\alpha\gamma\alpha^{-1}, \alpha\beta^{-1}\delta\gamma^{-1}\beta\gamma\alpha^{-1})}, \\ && \xi_{(\alpha, \beta)}^{(\gamma, \delta)}(p\bowtie h) = p\circ\beta\alpha^{-1} \bowtie \alpha\gamma^{-1}\beta^{-1}\gamma(h). \end{eqnarray*} $

Let $ H $ be a coFrobenius Hopf algebra with a left integral $ \varphi $, and $ t\in A $ is a cointegral in $ A $ such that $ \varphi(t) = 1 $. Recalling from [5] there is an element

$ \begin{eqnarray*} u\otimes v = : \sum t(\cdot \varphi_{(2)}) \otimes S^{-1}(\varphi_{(1)}) \in M(H\otimes \widehat{H}). \end{eqnarray*} $

Following from Lemma 9 in [5] we have:

● For any $ p\in \widehat{H} $ and $ h\in H $,

$ \begin{eqnarray} v\langle p, u\rangle = p, \qquad u\langle v, h\rangle = h. \end{eqnarray} $

(2.3)

● Let $ u\otimes v = u'\otimes v' $, then

$ \begin{eqnarray} (\Delta\otimes \iota)(u\otimes v) = u\otimes u'\otimes vv', \qquad (\iota\otimes\Delta)(u\otimes v) = uu'\otimes v\otimes v'. \end{eqnarray} $

(2.4)

And from [3] $ \mathcal{A} = \bigoplus_{(\alpha, \beta)\in G} \widehat{H} \bowtie H(\alpha, \beta) $ is a quasitriangular $ G $-cograded multiplier Hopf algebra with a generalized R-matrix given by

$ \begin{eqnarray*} R = \sum\limits_{(\alpha, \beta), (\gamma, \delta)\in G} R_{(\alpha, \beta), (\gamma, \delta)} = \sum\limits_{(\alpha, \beta), (\gamma, \delta)\in G} \varepsilon\bowtie\beta^{-1}(u) \otimes v\bowtie 1. \end{eqnarray*} $

Recalling from [2], a $ (\alpha, \beta) $-Yetter-Drinfel'd module over $ H $ is a vector space $ M $, such that $ M $ is a left $ H $-module (with notation $ h\otimes m \mapsto h\cdot m $) and a right $ H $-comodule (with notation $ M\rightarrow M\otimes H, m \mapsto m_{(0)}\otimes m_{(1)} $) with the following compatibility condition:

$ \begin{eqnarray} && (h\cdot m)_{(0)} \otimes (h\cdot m)_{(1)} = h_{(2)}\cdot m_{(0)} \otimes \beta(h_{(3)}) m_{(1)} \alpha S^{-1}(h_{(1)}). \end{eqnarray} $

(3.1)

We denote by $ {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $ the category of $ (\alpha, \beta) $-Yetter-Drinfel'd modules, morphism being the $ H $-linear $ H $-colinear maps. If $ H $ is "finite-dimensional", then

$ {}_{H}\mathcal{YD}^{H}(\alpha, \beta) \cong {}_{H^* \bowtie H(\alpha, \beta)}\mathcal{M}, $

where $ H^* \bowtie H(\alpha, \beta) $ is the diagonal crossed product.

One main question naturally arises: Does this isomorphism also hold for some "infinite-dimensional" Hopf algebras? In the following, we will give a positive answer to an infinite-dimensional coFrobenius Hopf algebra case.

Recalling from Lemma 11 in [5], If $ M $ is a left unital $ \widehat{H} $-module, then

$ \begin{eqnarray*} \rho:&& M\longrightarrow M\otimes H, \\ && m \mapsto \sum S^{-1}(\varphi_{(1)})\cdot m \otimes t(\cdot \varphi_{(2)}) = v\cdot m\otimes u \end{eqnarray*} $

gives the $ H $-comodule structure on $ M $. Following this lemma, we get the following proposition.

Proposition 3.1  If $ M \in {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M} $, then $ M \in {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $ with structures

$ \begin{eqnarray*} && h\cdot m = (\varepsilon \bowtie h) \cdot m, \\ && m\mapsto m_{(0)} \otimes m_{(1)} = (S^{-1}(\varphi_{(1)}) \bowtie 1)\cdot m \otimes t(\cdot \varphi_{(2)}). \end{eqnarray*} $

Proof  Here we treat $ \widehat{H} $ and $ H $ as subalgebras of $ \widehat{H} \bowtie H(\alpha, \beta) $ in the usual way, then it is easy to get $ M $ is an $ H $-module and $ H $-comodule.

To show $ M \in {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $, i.e., $ \rho(h\cdot m) = h_{(2)}\cdot m_{(0)} \otimes \beta(h_{(3)}) m_{(1)} \alpha S^{-1}(h_{(1)}) $, it is enough to verify that

$ \begin{eqnarray*} (S^{-1}(\varphi_{(1)}) \bowtie h) \otimes t(\cdot \varphi_{(2)}) = (\varepsilon \bowtie h_{(2)}) (S^{-1}(\varphi_{(1)}) \bowtie 1) \otimes \beta(h_{(3)}) t(\cdot \varphi_{(2)}) \alpha S^{-1}(h_{(1)}). \end{eqnarray*} $

Viewing $ \widehat{H} \bowtie H(\alpha, \beta) \otimes H $ as a subspace of $ Hom(\widehat{H}, (\widehat{H} \bowtie H)(\alpha, \beta)) $ in a natural way, we only need to check that

$ \begin{eqnarray*} p \bowtie h &\stackrel{(3)}{ = }& (S^{-1}(\varphi_{(1)}) \bowtie h) t(p \varphi_{(2)}) \\ & = & (\varepsilon \bowtie h_{(2)}) (S^{-1}(\varphi_{(1)}) \bowtie 1) \langle p, \beta(h_{(3)}) t(\cdot \varphi_{(2)}) \alpha S^{-1}(h_{(1)}) \rangle \end{eqnarray*} $

holds for any $ p\in \widehat{H} $. Indeed, for any $ p'\in \widehat{H} $,

$ \begin{eqnarray*} && (p' \bowtie h_{(2)}) (S^{-1}(\varphi_{(1)}) \bowtie 1) \langle p, \beta(h_{(3)}) t(\cdot \varphi_{(2)}) \alpha S^{-1}(h_{(1)}) \rangle \\ & = & (p' \bowtie h_{(2)}) (S^{-1}(\varphi_{(1)}) \bowtie 1) \langle p, \beta(h_{(3)}) t(\cdot \varphi_{(2)}) \alpha S^{-1}(h_{(1)}) \rangle \\ & = & (p' \bowtie h_{(2)}) (p_{(2)} \bowtie 1) \langle p_{(1)}, \beta(h_{(3)}) \rangle \langle p_{(3)}, \alpha S^{-1}(h_{(1)}) \rangle \\ & = & \langle p_{(1)}, \beta(h_{(5)}) \rangle \langle p_{(3)}, \alpha S^{-1}(h_{(1)}) \rangle \langle p_{(2)}, S^{-1}\beta(h_{(4)})\rangle \langle p_{(4)}, \alpha(h_{(1)})\rangle ( p'p_{(3)}\bowtie h_{(3)}) \\ & = & p'p\bowtie h. \end{eqnarray*} $

This completes the proof.

Proposition 3.2  If $ M \in {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $, then $ M \in {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M} $ with the structure

$ \begin{eqnarray*} (p\bowtie h) \cdot m = p((h \cdot m)_{(1)}) (h \cdot m)_{(0)}. \end{eqnarray*} $

Proof  It is straightforward to check that $ (p\bowtie h) \cdot \big((q\bowtie l) \cdot m \big) = \big( (p\bowtie h)(q\bowtie l)\big) \cdot m $. In fact,

$ \begin{eqnarray*} && (p\bowtie h) \cdot \big((q\bowtie l) \cdot m \big) \\ & = & (p\bowtie h) \cdot \big( q((l \cdot m)_{(1)}) (q \cdot l)_{(0)} \big) \\ & = & (p\bowtie h) \cdot \big( q(\beta(l_{(3)}) m_{(1)} \alpha S^{-1}(l_{(1)})) l_{(2)}\cdot m_{(0)} \big) \\ & = & q\big(\beta(l_{(3)}) m_{(1)} \alpha S^{-1}(l_{(1)})\big) p\big((hl_{(2)} \cdot m_{(0)})_{(1)}\big) (hl_{(2)} \cdot m_{(0)})_{(0)} \\ & = & q\big(\beta(l_{(5)}) m_{(2)} \alpha S^{-1}(l_{(1)})\big) p\big(\beta(h_{(3)}l_{(4)}) m_{(1)} \alpha S^{-1}(h_{(1)}l_{(2)})\big) (h_{(2)}l_{(3)}) \cdot m_{(0)}, \end{eqnarray*} $

and

$ \begin{eqnarray*} && \big( (p\bowtie h)(q\bowtie l)\big) \cdot m \\ & = & \big(\langle q_{(1)}, S^{-1}\beta(h_{(3)})\rangle \langle q_{(3)}, \alpha(h_{(1)})\rangle ( pq_{(2)}\bowtie h_{(2)}l)\big) \cdot m \\ & = & \langle q_{(1)}, S^{-1}\beta(h_{(3)})\rangle \langle q_{(3)}, \alpha(h_{(1)})\rangle (pq_{(2)})((h_{(2)}l \cdot m)_{(1)})\otimes (h_{(2)}l \cdot m)_{(0)} \\ & = & \langle q_{(1)}, S^{-1}\beta(h_{(5)})\rangle \langle q_{(3)}, \alpha(h_{(1)})\rangle (pq_{(2)})\big(\beta(h_{(4)}l_{(3)}) m_{(1)} \alpha S^{-1}(h_{(2)}l_{(1)}) \big) (h_{(3)}l_{(2)}) \cdot m_{(0)} \\ & = & \underline{\langle q_{(1)}, S^{-1}\beta(h_{(7)})\rangle \langle q_{(3)}, \alpha(h_{(1)})\rangle} \langle p, \beta(h_{(5)}l_{(4)}) m_{(1)} \alpha S^{-1}(h_{(3)}l_{(2)})\rangle \\ && \underline{\langle q_{(2)}, \beta(h_{(6)}l_{(5)}) m_{(2)} \alpha S^{-1}(h_{(2)}l_{(1)})\rangle } (h_{(4)}l_{(3)}) \cdot m_{(0)} \\ & = & \langle q, \beta(l_{(5)}) m_{(2)} \alpha S^{-1}(l_{(1)})\rangle \langle p, \beta(h_{(3)}l_{(4)}) m_{(1)} \alpha S^{-1}(h_{(1)}l_{(2)})\rangle (h_{(2)}l_{(3)}) \cdot m_{(0)}. \end{eqnarray*} $

Next, we get the main result of this section, generalizing the conclusion in [2] and giving an answer to the question introduced in Section 1.

Theorem 3.3  For a coFrobenius Hopf algebra $ H $,

$ \begin{eqnarray} {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M} \cong {}_{H}\mathcal{YD}^{H}(\alpha, \beta). \end{eqnarray} $

(3.2)

Proof  The correspondence easily follows from Proposition 3.1 and 3.2. Let $ f: M\rightarrow N $ be a morphism in $ {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $, i.e., $ f $ is a module and comodule map. Then in $ {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M} $,

$ \begin{eqnarray*} (f\otimes \iota)\rho(m) & = & f \big( (S^{-1}(\varphi_{(1)}) \bowtie 1)\cdot m \big) \otimes t(\cdot \varphi_{(2)}) \\ \rho(f(m)) & = & (S^{-1}(\varphi_{(1)}) \bowtie 1)\cdot f(m) \otimes t(\cdot \varphi_{(2)}). \end{eqnarray*} $

$ (f\otimes \iota)\rho(m) = \rho(f(m)) $ implies $ f $ is a $ \widehat{H}\bowtie 1 $-module map, and so a $ {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $-module map. We define a functor $ F_{(\alpha, \beta)}: {}_{H}\mathcal{YD}^{H}(\alpha, \beta) \longrightarrow {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M} $ as follows,

$ \begin{eqnarray*} F_{(\alpha, \beta)}(M) = M, \qquad \mbox{and} \qquad F_{(\alpha, \beta)}(f) = f. \end{eqnarray*} $

Conversely, if $ f: M\rightarrow N $ be a morphism in $ {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M} $, then

$ \begin{eqnarray*} (f\otimes \iota)\rho(m) & = & f \big( (S^{-1}(\varphi_{(1)}) \bowtie 1)\cdot m \big) \otimes t(\cdot \varphi_{(2)}) \\ & = & (S^{-1}(\varphi_{(1)}) \bowtie 1)\cdot f(m) \otimes t(\cdot \varphi_{(2)}) \\ & = & \rho(f(m)). \end{eqnarray*} $

This shows that $ f $ is a $ H $-comodule map. Then we similarly define a functor $ G_{(\alpha, \beta)}: {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M} \longrightarrow {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $ by $ G_{(\alpha, \beta)}(M) = M, \;\mbox{and} G_{(\alpha, \beta)}(f) = f. $

From above, $ F $ and $ G $ preserve the morphisms from each other. Also $ F_{(\alpha, \beta)}G_{(\alpha, \beta)} = 1_{(\alpha, \beta)} $ and $ G_{(\alpha, \beta)}F_{(\alpha, \beta)} = 1_{(\alpha, \beta)} $. We have established the equivalence between $ {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $ and $ {}_{\widehat{H} \bowtie H(\alpha, \beta)}\mathcal{M} $.

Corollary 3.4  Let $ H $ be a coFrobenius Hopf algebra and $ \alpha, \beta, \gamma \in Aut(H) $, then

$ \begin{eqnarray*} {}_{\widehat{H} \bowtie H(\alpha\beta, \gamma\beta)}\mathcal{M} \cong {}_{\widehat{H} \bowtie H(\alpha, \gamma)}\mathcal{M}. \end{eqnarray*} $

Proof  It follows straightforwardly from the fact $ {}_{H}\mathcal{YD}^{H}(\alpha\beta, \gamma\beta) \cong {}_{H}\mathcal{YD}^{H}(\alpha, \gamma) $.

Example 3.5  (1) When $ \alpha = \beta = \iota $, then $ \widehat{H} \bowtie H(\iota, \iota) = D(H) $ the quantum double of a coFrobenius Hopf algebra. Then we have the following result, which is the main result in [5], i.e., for a coFrobenius Hopf algebra $ H $,

$ {}_{H}\mathcal{YD}^{H} \cong {}_{\widehat{H} \bowtie H}\mathcal{M}. $

(2) When $ \alpha = S^{2} $ and $ \beta = \iota $, $ {}_{H}\mathcal{YD}^{H}(S^{2}, \iota) $ is exactly the category of anti-Yetter-Drinfeld modules defined in [9]. Then we have

$ \begin{eqnarray*} {}_{H}\mathcal{YD}^{H}(S^{2}, \iota) \cong {}_{\widehat{H} \bowtie H (S^{2}, \iota)}\mathcal{M}. \end{eqnarray*} $

Let $ \mathcal{YD}(H) $ be the disjoint union of $ {}_{H}\mathcal{YD}^{H}(\alpha, \beta) $ for every $ (\alpha, \beta)\in G $. Then following Section 3 in [2] or [4, 10] ($ H $ is a special multiplier Hopf algebra), we have that $ {}_{H}\mathcal{YD}^{H} $ is a braided $ T $-category with the structures as follows

● Tensor product: if $ V \in _{H}\mathcal{YD}^{H}(\alpha, \beta) $ and $ W \in _{H}\mathcal{YD}^{H}(\gamma, \delta) $ with $ \alpha, \beta , \gamma, \delta\in Aut(H) $, then $ V\otimes W \in _{H}\mathcal{YD}^{H} (\alpha\gamma, \delta\gamma^{-1}\beta\gamma) $, with the structures as follows:

$ \begin{eqnarray*} && h\cdot (v\otimes w) = \gamma(h_{(1)})\cdot v \otimes \gamma^{-1}\beta\gamma(h_{(2)})\cdot w, \\ && v\otimes w \mapsto (v\otimes w)_{(0)} \otimes (v\otimes w)_{(1)} = (v_{(0)} \otimes w_{(0)}) \otimes w_{(1)}v_{(1)} \end{eqnarray*} $

for all $ v\in V, w\in W $.

● Crossed functor: Let $ W\in _{H}\mathcal{YD}^{H}(\gamma, \delta) $, we define $ \xi_{(\alpha, \beta)}(W) = ^{(\alpha, \beta)}W = W $ as vector space, with structures: for all $ a, a'\in A $ and $ w\in W $

$ \begin{eqnarray*} && a\rightharpoonup w = \gamma^{-1} \beta\gamma\alpha^{-1} (a)\cdot w, \\ && w \mapsto w_{<0>} \otimes w_{<1>} = w_{(0)}\otimes \alpha\beta^{-1}(w_{(1)}). \end{eqnarray*} $

Then $ ^{(\alpha, \beta)}W \in _{H}\mathcal{YD}^{H}((\alpha, \beta)\# (\gamma, \delta)\# (\alpha, \beta)^{-1}) = _{H}\mathcal{YD}^{H}(\alpha\gamma\alpha^{-1}, \alpha\beta^{-1}\delta\gamma^{-1}\beta\gamma\alpha^{-1}) $.

The functor $ \xi_{(\alpha, \beta)} $ acts as identity on morphisms.

● Braiding: If $ V \in _{H}\mathcal{YD}^{H}(\alpha, \beta) $, and $ W \in _{H}\mathcal{YD}^{H}(\gamma, \delta) $. Taking $ ^{V}W = ^{(\alpha, \beta)}W $, we define a map $ C_{V, W}: V\otimes W \longrightarrow ^{V}W \otimes V $ by

$ \begin{eqnarray*} && C_{(\alpha, \beta), (\gamma, \delta)}(v\otimes w) = w_{(0)} \otimes \beta^{-1}(w_{(1)})\cdot v \end{eqnarray*} $

for all $ v\in V $ and $ w\in W $.

Following from Theorem 3.3, we obtain the following result, generalizing Theorem 3.10 in [2].

Theorem 3.6  For a coFrobenius Hopf algebra $ H $ and its $ G $-cograded multiplier Hopf algebra $ \mathcal{A} = \bigoplus_{(\alpha, \beta)\in G} \widehat{H} \bowtie H(\alpha, \beta) $, $ Rep(\mathcal{A}) $ and $ \mathcal{YD}(H) $ are isomorphic as braided $ T $-categories over $ G $.