---
abstract: 'We discuss properties of Yetter-Drinfeld modules over weak bialgebras over commutative rings. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If $H$ is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show that the category of Yetter-Drinfeld modules is isomorphic to the category of modules over the Drinfeld double. The category of finitely generated projective Yetter-Drinfeld modules over a weak Hopf algebra has duality.'
address:
- 'Faculty of Engineering Sciences, Vrije Universiteit Brussel, VUB, B-1050 Brussels, Belgium'
- 'Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China'
- 'Department of Mathematics, Shandong Institute of Architecture and Engineering, Jinan, Shandong 250014, China'
author:
- 'S. Caenepeel'
- Dingguo Wang
- Yanmin Yin
title: 'Yetter-Drinfeld modules over weak bialgebras'
---

[^1]

Introduction {#introduction .unnumbered}
============

Weak bialgebras and Hopf algebras are generalizations of ordinary bialgebras and Hopf algebras in the following sense: the defining axioms are the same, but the multiplicativity of the counit and comultiplicativity of the unit are replaced by weaker axioms. The easiest example of a weak Hopf algebra is a groupoid algebra; other examples are face algebras [@Hayashi], quantum groupoids [@Ocneanu], generalized Kac algebras [@Yamanouchi] and quantum transformation groupoids [@NV]. Temperley-Lieb algebras give rise to weak Hopf algebras (see [@NV]). A purely algebraic study of weak Hopf algebras has been presented in [@BohmNSI]. A survey of weak Hopf algebras and their applications may be found in [@NV]. It has turned out that many results of classical Hopf algebra theory can be generalized to weak Hopf algebras.\
Yetter-Drinfeld modules over finite dimensional weak Hopf algebras over fields have been introduced by Nenciu [@Nenciu]. It is shown in [@Nenciu] that the category of finite dimensional Yetter-Drinfeld modules is isomorphic to the category of finite dimensional modules over the Drinfeld double, as introduced in the appendix of [@Bohm]. It is also shown that this category is braided isomorphic to the center of the category of finite dimensional $H$-modules. In this note, we discuss Yetter-Drinfeld modules over weak bialgebras over commutative rings. The results in [@Nenciu] are slightly generalized and more properties are given.\
In [Section \[se:2\]]{}, we compute the weak center of the category of modules over a weak bialgebra $H$, and show that it is isomorphic to the category of Yetter-Drinfeld modules. If $H$ is a weak Hopf algebra, then the weak center equals the center. In this situation, properties of the center construction can be applied to show that the four categories of Yetter-Drinfeld modules, namely the left-left, left-right, right-left and right-right versions, are isomorphic as braided monoidal categories. Here we apply methods that have been used before in [@BCP], in the case of quasi-Hopf algebras.\
In [@cmz], it was observed that Yetter-Drinfeld modules over a classical Hopf algebra are special cases of Doi-Hopf modules, as introduced by Doi and Koppinen (see [@Doi92; @Koppinen95]). In [Section \[se:3\]]{}, we will show that Yetter-Drinfeld modules over weak Hopf algebras are weak Doi-Hopf modules, in the sense of Böhm [@Bohm], and, a fortiori, weak entwined modules [@CDG], and comodules over a coring [@BW].\
The advantage of this approach is that it leads easily to a new description of the Drinfeld double of a finitely generated projective weak Hopf algebra, using methods developed in [@CDG]: we define the Drinfeld double as a weak smash product of $H$ and its dual. We show that our Drinfeld double is equal to the Drinfeld double of [@Bohm; @Nenciu] (see [Proposition \[pr:4.2\]]{}) and anti-isomorphic to the Drinfeld double of [@NVT] (see [Proposition \[pr:4.4\]]{}). In [Section \[se:5\]]{}, we show that the category of finitely generated projective Yetter-Drinfeld modules over a weak Hopf algebra has duality.\
In Sections \[se:1.1\] and \[se:1.2\], we recall some general properties of weak bialgebras and Hopf algebras. Further detail can be found in [@BW; @BohmNSI; @NV]. In [Section \[se:1.3\]]{}, we recall the center construction, and in [Section \[se:1.4\]]{}, we recall the notions of weak Doi-Hopf modules, weak entwining structures and weak smash products.

Preliminary results
===================

[\[se:1\]]{}

Weak bialgebras
---------------

[\[se:1.1\]]{} Let $k$ be a commutative ring. Recall that a weak $k$-bialgebra is a $k$-module with a $k$-algebra structure $(\mu,\eta)$ and a $k$-coalgebra structure $(\Delta,\varepsilon)$ such that $\Delta(hk)=\Delta(h)\Delta(k)$, for all $h,k\in H$, and $$\begin{aligned}
\Delta^2(1)&=& 1_{(1)}\ot 1_{(2)}1_{(1')}\ot 1_{(2')}=
1_{(1)}\ot 1_{(1')}1_{(2)}\ot 1_{(2')},{\label{eq:1.1.1}}\\
\varepsilon(hkl)&=& \varepsilon(hk_{(1)})\varepsilon(k_{(2)}l)=
\varepsilon(hk_{(2)})\varepsilon(k_{(1)}l),{\label{eq:1.1.2}}\end{aligned}$$ for all $h,k,l\in H$. We use the Sweedler-Heyneman notation for the comultiplication, namely $$\Delta(h)=h_{(1)}\ot h_{(2)}=h_{(1')}\ot h_{(2')}.$$ We summarize the elementary properties of weak bialgebras. The proofs are direct applications of the defining axioms (see [@BohmNSI; @NV]). We have idempotent maps $\varepsilon_t,~\varepsilon_s:\ H\to H$ defined by $$\varepsilon_t(h)=\varepsilon(1_{(1)}h)1_{(2)}~~;
\varepsilon_s(h)=1_{(1)}\varepsilon(h1_{(2)}).$$ $\varepsilon_t$ and $\varepsilon_s$ are called the target map and the source map, and their images $H_t={{\rm Im}\,}(\varepsilon_t)={{\rm Ker}\,}(H-\varepsilon_t)$ and $H_s={{\rm Im}\,}(\varepsilon_s)={{\rm Ker}\,}(H-\varepsilon_s)$ are called the target and source space. For all $g,h\in H$, we have $${\label{eq:1.1.3}}
h_{(1)}\ot \varepsilon_t(h_{(2)})=1_{(1)}h\ot 1_{(2)}~~{\rm and}~~
\varepsilon_s(h_{(1)})\ot h_{(2)}=1_{(1)}\ot h1_{(2)},$$ and $${\label{eq:1.1.3*}}
h\varepsilon_t(g)=\varepsilon(h_{(1)}g)h_{(2)}~~{\rm and}~~
\varepsilon_s(g)h=h_{(1)}\varepsilon(gh_{(2)}).$$ From [(\[eq:1.1.3\*\])]{}, it follows immediately that $${\label{eq:1.1.3a}}
\varepsilon(h\varepsilon_t(g))=\varepsilon(hg)~~{\rm and}~~
\varepsilon(\varepsilon_s(g)h)=\varepsilon(gh).$$ The source and target space can be described as follows: $$\begin{aligned}
H_t=\{h\in H~|~\Delta(h)=1_{(1)}h\ot 1_{(2)}\}=
\{\phi(1_{(1)})1_{(2)}~|~\phi\in H^*\};{\label{eq:1.1.4}}\\
H_s=\{h\in H~|~\Delta(h)=1_{(1)}\ot h1_{(2)}\}=
\{1_{(1)}\phi(1_{(2)})~|~\phi\in H^*\}.{\label{eq:1.1.5}}\end{aligned}$$ We also have $${\label{eq:1.1.6}}
\varepsilon_t(h)\varepsilon_s(k)=\varepsilon_s(k)\varepsilon_t(h),$$ and its dual property $${\label{eq:1.1.6*}}
\varepsilon_s(h_{(1)})\ot \varepsilon_t(h_{(2)})=
\varepsilon_s(h_{(2)})\ot \varepsilon_t(h_{(1)}).$$ Finally $\varepsilon_s(1)=\varepsilon_t(1)=1$, and $${\label{eq:1.1.7}}
\varepsilon_t(h)\varepsilon_t(g)=\varepsilon_t(\varepsilon_t(h)g)~~{\rm and}~~
\varepsilon_s(h)\varepsilon_s(g)=\varepsilon_s(h\varepsilon_s(g)).$$ This implies that $H_s$ and $H_t$ are subalgebras of $H$.\

[\[le:1.2\]]{} Let $H$ be a weak bialgebra over a commutative ring. Then $\Delta(1)\in H_s\ot H_t$.

Applying $H\ot \varepsilon\ot H$ to [(\[eq:1.1.1\])]{}, we find that $1_{(1)}\ot 1_{(2)}=\varepsilon_s(1_{(1)})\ot 1_{(2)}
\in H_s\ot H$ and $1_{(1)}\ot 1_{(2)}=1_{(1)}\ot \varepsilon_t(1_{(2)})\in H\ot H_t$. Now let $K_s={{\rm Ker}\,}(\varepsilon_s)$, $K_t={{\rm Ker}\,}(\varepsilon_t)$. Then $H= H_s\oplus K_s= H_t\oplus K_t,$ and $$H\ot H=H_s\ot H_t\oplus H_s\ot K_t \oplus K_s\ot H_t \oplus K_s\ot K_t,$$ so it follows that $H_s\ot H_t= H\ot H_t \cap H_s\ot H$.

The target and source map for the weak bialgebra $H^{\rm op}$ are $${\label{eq:1.2a.1}}
\ol{\varepsilon}_t(h)=\varepsilon(h1_{(1)})1_{(2)}\in H_t~~{\rm and}~~
\ol{\varepsilon}_s(h)=\varepsilon(1_{(2)}h)1_{(1)}\in H_s.$$ $\ol{\varepsilon}_t$ and $\ol{\varepsilon}_s$ are also projections.\
The source and target space are anti-isomorphic, and they are separable Frobenius algebras over $k$. This was first proved for weak Hopf algebras (see [@BohmNSI]), and then generalized to weak bialgebras (see [@Schauenburg]).

[\[eq:1.2a\]]{} [@Schauenburg] Let $H$ be a weak bialgebra. Then $\ol{\varepsilon}_s$ restricts to an anti-algebra isomorphism $H_t\to H_s$ with inverse $\varepsilon_t$, and $\ol{\varepsilon}_t$ restricts to an anti-algebra isomorphism $H_s\to H_t$ with inverse $\varepsilon_s$.

[\[pr:1.2b\]]{} [@Schauenburg] Let $H$ be a weak bialgebra. Then $H_s$ and $H_t$ are Frobenius separable $k$-algebras. The separability idempotents of $H_t$ and $H_s$ are $$e_t=\varepsilon_t(1_{(1)})\ot 1_{(2)}=1_{(2)}\ot \ol{\varepsilon}_t(1_{(1)});$$ $$e_s=1_{(1)}\ot \varepsilon_s(1_{(2)})=\ol{\varepsilon}_s(1_{(2)})\ot 1_{(1)}.$$ The Frobenius systems for $H_t$ and $H_s$ are respectively $(e_t,\varepsilon_{|H_t})$ and $(e_s, \varepsilon_{|H_s})$. In particular, we have for all $z\in H_t$ that $${\label{eq:1.2b.1}}
z\varepsilon_t(1_{(1)})\ot 1_{(2)}=
\varepsilon_t(1_{(1)})\ot 1_{(2)}z.$$

It was shown in [@NVT] that the category of modules over a weak Hopf algebra is monoidal; it follows from the results of [@Schauenburg] that this property can be generalized to weak bialgebras. We explain now how this can be done directly.\
Let $M$ be a left $H$-module. By restriction of scalars, $M$ is a left $H_t$-module; $M$ becomes an $H_t$-bimodule, if we define a right $H_t$-action by $$m\cdot z=\ol{\varepsilon}_s(z)m.$$\
Let $M,N\in {}_H{\mathcal{M}}$, the category of left $H$-modules. We define $$M\ot_t N=\Delta(1)(M\ot N),$$ the $k$-submodule of $M\ot N$ generated by elements of the form $1_{(1)}\ot 1_{(2)}$. $M\ot_t N$ is a left $H$-module, with left diagonal action $h\cdot(m\ot n)=h_{(1)}m\ot h_{(2)}n$. It follows from [(\[eq:1.1.1\])]{} that the tensor product $\ot_t$ is associative. Observe that $$M\ot_t N\ot_t P=\Delta^2(1)(M\ot N\ot P).$$ $H_t\in {}_H{\mathcal{M}}$, with left $H$-action $h\leftact z=\varepsilon_t(hz)$. The induced $H_t$-bimodule structure is given by left and right multiplication by elements of $H_t$.\
For $M,N\in {}_H{\mathcal{M}}$, consider the projection $$\pi:\ M\ot N\to M\ot_t N,~~\pi(m\ot n)=1_{(1)}m\ot 1_{(2)}n.$$ Applying $\ol{\varepsilon}_s\ot H_t$ to [(\[eq:1.2b.1\])]{}, we find $$\ol{\varepsilon}_s(z\varepsilon_t(1_{(1)}))\ot 1_{(2)}=
1_{(1)}\ol{\varepsilon}_s(z)\ot 1_{(2)}=1_{(1)}\ot 1_{(2)}z,$$ hence $$\pi(mz\ot n)=\pi(\ol{\varepsilon}_s(z)m\ot n)=
1_{(1)}\ol{\varepsilon}_s(z)m\ot 1_{(2)}n
=1_{(1)}m\ot 1_{(2)}zn=\pi(m\ot zn).$$ So $\pi$ induces a map $\ol{\pi}:\ M\ot_{H_t} N\to M\ot_t N$, which is a left $H_t$-module isomorphism with inverse given by $$\ol{\pi}^{-1}(1_{(1)}m\ot 1_{(2)}n)=1_{(1)}m\ot_{H_t} 1_{(2)}n=m\ot_{H_t}n.$$

[\[pr:1.2c\]]{} Let $H$ be a weak bialgebra. Then we have a monoidal category $({}_H{\mathcal{M}},\ot_t,H_t,a,l,r)$. The associativity constraints are the natural ones. The left and right unit constraints $l_M:\ H_t\ot_t M\to M$ and $r_M:\ M\ot_t H_t\to M$ and their inverses are given by the formulas $$l_M(1_{(1)}\leftact z\ot 1_{(2)}m)=zm~~;~~l_M^{-1}(m)=\varepsilon_t(1_{(1)})\ot 1_{(2)}m;$$ $$r_M(1_{(1)}m\ot 1_{(2)}\leftact z)=\ol{\varepsilon}_s(z)m~~;~~
r_M^{-1}(m)=1_{(1)}m\ot 1_{(2)}.$$

This is a direct consequence of the observations made above. Let us check that $$\begin{aligned}
&&\hspace*{-2cm}
l_M^{-1}(l_M(1_{(1)}\leftact z\ot 1_{(2)}m))=l_M^{-1}(zm)=
\varepsilon_t(1_{(1)})\ot 1_{(2)}zm\\
&=&z\varepsilon_t(1_{(1)})\ot 1_{(2)}m\equal{{(\ref{eq:1.1.7})}}\varepsilon_t(z1_{(1)})\ot 1_{(2)}m\\
&=&\varepsilon_t(1_{(1)}z)\ot 1_{(2)}m=1_{(1)}\leftact z\ot 1_{(2)}m\\
&&\hspace*{-2cm}
l_M(l_M^{-1}(m))=l_M(\varepsilon_t(1_{(1)})\ot 1_{(2)}m)=m\\
&&\hspace*{-2cm}
r_M^{-1}(r_M(1_{(1)}m\ot 1_{(2)}\leftact z))=r_M^{-1}(\ol{\varepsilon}_s(z)m)=
1_{(1)}\ol{\varepsilon}_s(z)m\ot 1_{(2)}\\
&=& 1_{(1)}m\ot 1_{(2)}z=1_{(1)}m\ot 1_{(2)}\leftact z\\
&&\hspace*{-2cm}
r_M(r_M^{-1}(m))=r_M(1_{(1)}m\ot 1_{(2)})=\ol{\varepsilon}_s(1)m=m.\end{aligned}$$

Weak Hopf algebras
------------------

[\[se:1.2\]]{} A weak Hopf algebra is a weak bialgebra together with a map $S:\ H\to H$, called the antipode, satisfying $${\label{eq:1.3.1}}
S*H=\varepsilon_s,~~H*S=\varepsilon_t,~~{\rm and}~~S*H*S=S,$$ where $*$ is the convolution product. It follows immediately that $${\label{eq:1.3.2}}
S=\varepsilon_s*S=S*\varepsilon_t.$$ If the antipode exists, then it is unique. We will always assume that $S$ is bijective; if $H$ is a finite dimensional weak Hopf algebra over a field, then $S$ is automatically bijective (see [@BohmNSI Theorem 2.10]).

[\[le:1.5\]]{} Let $H$ be a weak Hopf algebra. Then $S$ is an anti-algebra and an anti-coalgebra morphism. For all $h,g\in H$, we have $$\begin{aligned}
\varepsilon_t(hg)&=&
\varepsilon_t(h\varepsilon_t(g))=h_{(1)}\varepsilon_t(g)S(h_{(2)});{\label{eq:1.4.1}}\\
\varepsilon_s(hg)&=&
\varepsilon_s(\varepsilon_s(h)g)=S(g_{(1)})\varepsilon_s(h)g_{(2)};
{\label{eq:1.4.1a}}\\
\Delta(\varepsilon_t(h))&=&h_{(1)}S(h_{(3)})\ot\varepsilon_t(h_{(2)}){\label{eq:1.4.2}}\\
\Delta(\varepsilon_s(h))&=&\varepsilon_s(h_{(2)})\ot S(h_{(1)})h_{(3)}{\label{eq:1.4.2a}}.\end{aligned}$$

[\[le:1.6\]]{} Let $H$ be a weak Hopf algebra. For all $h\in H$, we have $$\begin{aligned}
\varepsilon_t(h)&=&\varepsilon(S(h)1_{(1)})1_{(2)}=
\varepsilon(1_{(2)}h)S(1_{(1)})=S(\ol{\varepsilon}_s(h)){\label{eq:1.6.1}}\\
\varepsilon_s(h)&=&1_{(1)}\varepsilon(1_{(2)}S(h))=
\varepsilon(h1_{(1)})S(1_{(2)})=S(\ol{\varepsilon}_t(h)){\label{eq:1.6.2}}.\end{aligned}$$

[\[co:1.6b\]]{} Let $H$ be a weak Hopf algebra. For all $h\in H$, we have $${\label{eq:1.6.3}}
\varepsilon_t(h_{(1)})\ot h_{(2)}=S(1_{(1)})\ot 1_{(2)}h~~;~~
h_{(1)}\ot \varepsilon_s(h_{(2)})=h1_{(1)}\ot S(1_{(2)}).$$

[\[pr:1.7\]]{} Let $H$ be a weak Hopf algebra. Then $${\label{eq:1.7.1}}
\varepsilon_t\circ S=\varepsilon_t\circ \varepsilon_s= S\circ \varepsilon_s~~;~~
\varepsilon_s\circ S=\varepsilon_s\circ \varepsilon_t= S\circ \varepsilon_t.$$

[\[co:1.8\]]{} Let $H$ be a weak Hopf algebra with bijective antipode. Then $S_{|H_t}=({\varepsilon}_s)_{|H_t}$, and $S^{-1}_{|H_s}=(\ol{\varepsilon}_t)_{|H_s}$, so $S$ restricts to an anti-algebra isomorphism $H_t\to H_s$.

It follows that the separability idempotents of $H_t$ and $H_s$ are $e_t=S(1_{(1)})\ot 1_{(2)}$ and $e_s=1_{(1)}\ot S(1_{(2)})$. Consequently, we have the following formulas, for $z\in H_t$ and $y\in H_s$: $$\begin{aligned}
zS(1_{(1)})\ot 1_{(2)}&=&S(1_{(1)})\ot 1_{(2)}z;{\label{eq:1.9.3}}\\
y1_{(1)}\ot 1_{(2)}&=& 1_{(1)}\ot S^{-1}(y)1_{(2)}.{\label{eq:1.9.4}}\end{aligned}$$ Applying $S^{-1}\ot H$ to [(\[eq:1.9.3\])]{}, we find $${\label{eq:1.9.2}}
1_{(1)}S^{-1}(z)\ot 1_{(2)}=1_{(1)}\ot 1_{(2)}z.$$

The center of a monoidal category
---------------------------------

[\[se:1.3\]]{} Let ${\mathcal{C}}=({\mathcal{C}},\ot,I,a,l,r)$ be a monoidal category. The weak left center ${\mathcal{W}}_l({\mathcal{C}})$ is the category with the following objects and morphisms. An object is a couple $(M,\sigma_{M, -})$, with $M\in {\mathcal{C}}$ and $\sigma_{M, -}:\ M\ot -\to -\ot M$ a natural transformation, satisfying the following condition, for all $X,Y\in {\mathcal{C}}$: $${\label{eq:center1}}
(X\ot \sigma_{M, Y})\circ a_{X,M,Y}\circ (\sigma_{M, X}\ot Y)=a_{X,Y,M}\circ \sigma_{M, X\ot Y}
\circ a_{M,X,Y},$$ and such that $\sigma_{M, I}$ is the composition of the natural isomorphisms $M\ot I\cong M\cong I\ot M$. A morphism between $(M,\sigma_{M, -})$ and $(M',\sigma_{M', -})$ consists of $\vartheta:\ M\to M'$ in ${\mathcal{C}}$ such that $$(X\ot \vartheta)\circ \sigma_{M, X}=\sigma_{M', X}\circ (\vartheta\ot X).$$ The left center ${\mathcal{Z}}_l({\mathcal{C}})$ is the full subcategory of ${\mathcal{W}}_l({\mathcal{C}})$ consisting of objects $(M,\sigma_{M, -})$ with $\sigma_{M,-}$ a natural isomorphism. ${\mathcal{Z}}_l({\mathcal{C}})$ is a braided monoidal category. The tensor product is $$(M,\sigma_{M, -})\ot (M',\sigma_{M', -})=(M\ot M', \sigma_{M\ot M', -})$$ with $${\label{eq:center2}}
\sigma_{M\ot M', X}=a_{X,M,M'}\circ (\sigma_{M, X}\ot M')\circ a^{-1}_{M,X,M'}
\circ (M\ot \sigma_{M', X})\circ a_{M,M',X},$$ and the unit is $(I,\sigma_{I,-})$, with $${\label{eq:center2bis}}
\sigma_{I,M}=r_M^{-1}\circ l_M.$$ The braiding $c$ on ${\mathcal{Z}}_l({\mathcal{C}})$ is given by $${\label{eq:center3}}
c_{M,M'}=\sigma_{M, M'}:\ (M, \sigma_{M, -})\ot (M', \sigma_{M', -})
\to (M',\sigma_{M', -})\ot (M,\sigma_{M, -}).$$ ${\mathcal{Z}}_l({\mathcal{C}})^{\rm in}$ will be our notation for the monoidal category ${\mathcal{Z}}_l({\mathcal{C}})$, together with the inverse braiding $\tilde{c}$ given by $\tilde{c}_{M,M'}=c^{-1}_{M',M}=\sigma^{-1}_{M', M}$.

The right center ${\mathcal{Z}}_r({\mathcal{C}})$ is defined in a similar way. An object is a couple $(M,\tau_{-, M})$, where $M\in {\mathcal{C}}$ and $\tau_{-, M}:\ -\ot M\to M\ot -$ is a family of natural isomorphisms such that $\tau_{-, I}$ is the natural isomorphism and $${\label{eq:center4}}
a^{-1}_{M, X, Y}\circ \tau_{X\ot Y, M}\circ a^{-1}_{X, Y, M}=(\tau_{X, M}\ot Y)
\circ a^{-1}_{X, M, Y}\circ (X\ot \tau_{Y, M}),$$ for all $X,Y\in {\mathcal{C}}$. A morphism between $(M,\tau_{-, M})$ and $(M',\tau_{-, M'})$ consists of $\vartheta:\ M\to M'$ in ${\mathcal{C}}$ such that $$(\vartheta\ot X)\circ \tau_{X, M}=\tau_{X, M'}\circ (X\ot \vartheta),$$ for all $X\in {\mathcal{C}}$. ${\mathcal{Z}}_r({\mathcal{C}})$ is a braided monoidal category. The unit is $(I, l_{-}^{-1}\circ r_{-})$ and the tensor product is $$(M,\tau_{-, M})\ot (M',\tau_{-, M'})=(M\ot M', \tau_{-, M\ot M'})$$ with $${\label{eq:center5}}
\tau_{X, M\ot M'}=a^{-1}_{M,M',X}\circ (M\ot \tau_{X, M'})\circ a_{M,X,M'}
\circ (\tau_{X, M}\ot M')\circ a^{-1}_{X,M,M'}.$$ The braiding $d$ is given by $${\label{eq:center6}}
d_{M,M'}=\tau_{M, M'}:\ (M, \tau_{-, M})\ot (M', \tau_{-, M'})\to 
(M', \tau_{-, M'})\ot (M, \tau_{-, M}).$$ ${\mathcal{Z}}_r({\mathcal{C}})^{\rm in}$ is the monoidal category ${\mathcal{Z}}_r({\mathcal{C}})$ with the inverse braiding $\tilde{d}$ given by $\tilde{d}_{M,M'}=d^{-1}_{M',M}=\tau^{-1}_{M', M}$.

For details in the case where ${\mathcal{C}}$ is a strict monoidal category, we refer to [@Kassel Theorem XIII.4.2]. The results remain valid in the case of an arbitrary monoidal category, since every monoidal category is equivalent to a strict one. Recall the following result from [@BCP].

[\[pr:1.0.1\]]{} Let ${\mathcal{C}}$ be a monoidal category. Then we have an isomorphism of braided monoidal categories $F:\ {\mathcal{Z}}_l({\mathcal{C}})\to {\mathcal{Z}}_r({\mathcal{C}})^{\rm in}$, given by $$F(M, \sigma_{M, -})=(M, \sigma^{-1}_{M, -})~~{\rm and}~~F(\vartheta)=\vartheta .$$

We have a second monoidal structure on ${\mathcal{C}}$, defined as follows: $$\ol{{\mathcal{C}}}=({\mathcal{C}},\ol{\ot}=\ot\circ\tau,I, \ol{a}, r, l)$$ with $\tau:\ {\mathcal{C}}\times{\mathcal{C}}\to {\mathcal{C}}\times {\mathcal{C}}$, $\tau(M,N)=(N,M)$ and $\ol{a}$ defined by $\ol{a}_{M,N,X}=a^{-1}_{X,N,M}$.

If $c$ is a braiding on ${\mathcal{C}}$, then $\ol{c}$, given by $\ol{c}_{M,N}=
c_{N,M}$ is a braiding on $\ol{{\mathcal{C}}}$. In [@BCP], the following obvious result was stated.

[\[pr:1.0.2\]]{} Let ${\mathcal{C}}$ be a monoidal category. Then $$\ol{{\mathcal{Z}}_l({\mathcal{C}})}\cong {\mathcal{Z}}_r(\ol{{\mathcal{C}}})~~;~~
\ol{{\mathcal{Z}}_r({\mathcal{C}})}\cong {\mathcal{Z}}_l(\ol{{\mathcal{C}}})$$ as braided monoidal categories.

Weak entwining structures and weak smash products
-------------------------------------------------

[\[se:1.4\]]{} The results in this Section are taken from [@CDG]. Let $A$ be a ring without unit. $e\in A$ is called a preunit if $ea=ae=ae^2$, for all $a\in A$. Then map $p:\ A\to A$, $p(a)=ae$, satisfies the following properties: $p\circ p=p$ and $p(ab)=p(a)p(b)$. Then $\ol{A}={\rm Coim}(p)$ is a ring with unit $\ol{e}$ and $\ul{A}={{\rm Im}\,}(p)$ is a ring with unit $e^2$. $p$ induces a ring isomorphism $\ol{A}\to \ul{A}$.\
Let $k$ be a commutative ring, $A$, $B$ $k$-algebras with unit, and $R:\ B\ot A\to A\ot B$ a $k$-linear map. We use the notation $${\label{eq:smash}}
R(b\ot a)=a_R\ot b_R=a_r\ot b_r,$$ where the summation is implicitely understood. $A\#_RB$ is the $k$-algebra $A\ot B$ with newly defined multiplication $$(a\# b)(c\# d)= ac_R\# b_Rd.$$ $(A,B,R)$ is called a weak smash product structure if $A\#_RB$ is an associative $k$-algebra with preunit $1_A\# 1_B$. The multiplication is associative if and only if $$R(bd\ot a)=a_{Rr}\ot b_rd_R~~{\rm and}~~
R(b\ot ac)=a_Rc_r\ot b_{Rr},$$ for all $a,c\in A$ and $b,d\in B$. $1_A\# 1_B$ is a preunit if and only if $$R(1_B\ot a)=a(1_A)_R\ot (1_B)_R~~{\rm and}~~R(b\ot 1_A)=(1_A)_R\ot (1_B)_Rb.$$

A left-right weak entwining structure is a triple $(A,C,\psi)$, where $A$ is an algebra, $C$ is a coalgebra, and $\psi:\ A\ot C\to A\ot C$ is a $k$-linear map satisfying the conditions $$a_\psi\ot \Delta(c^\psi)=a_{\psi\Psi}\ot c_{(1)}^\Psi\ot c_{(2)}^\psi~~;~~
(ab)_\psi\ot c^\psi=a_\psi b_\Psi\ot c^{\Psi\psi};$$ $$1_\psi\ot c^\psi=\varepsilon(c_{(1)}^\psi)1_\psi\ot c_{(2)}~~;~~
a_\psi\varepsilon(c^\psi)=\varepsilon(c^\psi)a1_\psi.$$ Here we use the notation (with summation implicitely understood): $$\psi(a\ot c)=a_\psi\ot c^\psi.$$ An entwined module is a $k$-module $M$ with a left $A$-action and a right $C$-coaction such that $$\rho(am)=a_\psi m_{[0]}\ot m_{[1]}^\psi.$$ The category of entwined modules and left $A$-linear right $C$-colinear maps is denoted by ${}_A{\mathcal{M}}(\psi)^C$.\
Let $H$ be a weak bialgebra, and $A$ a right $H$-comodule, which is also an algebra with unit. $A$ is called a right $H$-comodule algebra if $\rho(a)\rho(b)=\rho(ab)$ and $1_{[0]}\ot \varepsilon_t(1_{[1]})=\rho(1)$.\
From [@Bohm], we recall the following definitions. Let $C$ be a left $H$-module which is also a coalgebra with counit. $C$ is called a left $H$-comodule algebra if $\Delta_C(hc)= \Delta_H(h)\Delta_C(c)$ and $${\label{eq:3.1.1}}
\varepsilon_C(hkc)=\varepsilon_H(hk_{(2)})\varepsilon_C(k_{(1)}c),$$ for all $c\in C$ and $h,k\in H$. Several equivalent definitions are given in [@CDG Sec. 4]. We then call $(H,A,C)$ a left-right weak Doi-Hopf datum. A weak Doi-Hopf module over $(H,A,C)$ is a $k$-module $M$ with a left $A$-action and a right $C$-coaction, satisfying the following compatibility relation, for all $m\in M$ and $a\in A$: $${\label{eq:3.1.2}}
\rho(am)=a_{[0]}m_{[0]}\ot a_{[1]}m_{[1]}.$$ The category of weak Doi-Hopf modules over $(H,A,C)$ and left $A$-linear right $C$-colinear maps is denoted by ${}_A{\mathcal{M}}(H)^C$.\
Let $(H,A,C)$ be a weak left-right Doi-Hopf datum, and consider the map $$\psi:\ A\ot C\to A\ot C,~~\psi(a\ot c)=a_{[0]}\ot a_{[1]}c.$$ Then $(A,C,\psi)$ is a weak left-right entwining structure, and we have an isomorphism of categories ${}_A{\mathcal{M}}(H)^C\cong {}_A{\mathcal{M}}(\psi)^C$.\
Let $(A,C,\psi)$ be a weak left-right entwining structure, and assume that $C$ is finitely generated projective as a $k$-module, with finite dual basis $\{(c_i,c_i^*)~|~i=1,\cdots,n\}$. Then we have a weak smash product structure $(A,C^*,R)$, with $R:\ C^*\ot A\to A\ot C^*$ given by $${\label{eq:1.41}}
R(c^*\ot a)=\hbox{$\sum_i$} \lan c^*,c_i^\psi\ran a_\psi\ot c_i^*.$$ We have an isomorphism of categories $${\label{eq:1.42}}
F:\ {}_A{\mathcal{M}}(\psi)^C\to {}_{\ol{A\#_R C^*}}{\mathcal{M}},$$ defined also follows: $F(M)=M$ as a $k$-module, with action $[a\# c^*]\cdot m=\lan c^*,m_{[1]}\ran am_{[0]}.$ Details can be found in [@CDG Theorem 3.4].

Yetter-Drinfeld modules over weak Hopf algebras
===============================================

[\[se:2\]]{} Let $H$ be a weak bialgebra. A left-left Yetter-Drinfeld module is a $k$-module with a left $H$-action and a left $H$-coaction such that the following conditions hold, for all $m\in M$ and $h\in H$: $$\begin{aligned}
&&\hspace*{-2cm}\lambda(m)= m_{[-1]}\ot m_{[0]}\in H\ot_t M;{\label{eq:2.1.1}}\\
&&\hspace*{-2cm} h_{(1)}m_{[-1]}\ot h_{(2)}m_{[0]}=(h_{(1)}m)_{[-1]}h_{(2)}\ot 
(h_{(1)}m)_{[0]}.{\label{eq:2.1.2}}\end{aligned}$$ We will now state some equivalent definitions. First we will rewrite the counit property for Yetter-Drinfeld modules.

[\[le:2.1\]]{} Let $H$ be a weak bialgebra, and $\lambda:\ M\to H\ot_t M$, $\rho(m)=m_{[-1]}\ot m_{[0]}$ a $k$-linear map. Then $${\label{eq:2.1.3}}
\varepsilon(m_{[-1]})m_{[0]}=\varepsilon_t(m_{[-1]})m_{[0]}.$$ Consequently, in the definition of a Yetter-Drinfeld module, the counit property $\varepsilon(m_{[-1]})m_{[0]}=m$ can be replaced by $\varepsilon_t(m_{[-1]})m_{[0]}=m$.

$$\varepsilon_t(m_{[-1]})m_{[0]}=\varepsilon(1_{(1)}m_{[-1]})1_{(2)}m_{[0]}=
\varepsilon(m_{[-1]})m_{[0]}.$$

In the case of a weak Hopf algebra, the compatibility relation [(\[eq:2.1.2\])]{} can also be restated:

[\[pr:2.2\]]{} (cf. [@Nenciu Remark 2.6]) Let $H$ be a weak Hopf algebra, and $M$ a $k$-module, with a left $H$-action and a left $H$-coaction. $M$ is a Yetter-Drinfeld module if and only if $${\label{eq:2.2.1}}
\lambda(hm)=h_{(1)}m_{[-1]}S(h_{(3)})\ot h_{(2)}m_{[0]}.$$

Let $M$ be a Yetter-Drinfeld module. Then we compute $$\begin{aligned}
&&\hspace*{-15mm}
h_{(1)}m_{[-1]}S(h_{(3)})\ot h_{(2)}m_{[0]}=
(h_{(1)}m)_{[-1]}h_{(2)}S(h_{(3)})\ot  (h_{(1)}m)_{[0]}\\
&=& (h_{(1)}m)_{[-1]}\varepsilon_t(h_{(2)})\ot  (h_{(1)}m)_{[0]}
\equal{{(\ref{eq:1.1.3})}}
(1_{(1)}hm)_{[-1]}1_{(2)}\ot  (1_{(1)}hm)_{[0]}\\
&\equal{{(\ref{eq:2.1.2})}}& 1_{(1)}(hm)_{[-1]}\ot
1_{(2)}(hm)_{[0]}\equal{{(\ref{eq:2.1.1})}}
(hm)_{[-1]}\ot (hm)_{[0]}=\lambda(hm).\end{aligned}$$ Conversely, assume that [(\[eq:2.2.1\])]{} holds for all $h\in H$ and $m\in M$. Taking $h=1$ in [(\[eq:2.2.1\])]{}, we find $$\begin{aligned}
\lambda(m)&=&1_{(1)}m_{[-1]}S(1_{(3)})\ot 1_{(2)}m_{[0]}\\
&=& 1_{(1)}m_{[-1]}S(1_{(2')})\ot 1_{(2)}1_{(1')}m_{[0]}\in H\ot_t M\end{aligned}$$ and $$\begin{aligned}
\lambda(m)&=& 1_{(1)}m_{[-1]}S(1_{(3)})\ot 1_{(2)}m_{[0]}
= 1_{(1)}m_{[-1]}S(1_{(2')})\ot 1_{(1')}1_{(2)}m_{[0]}\nonumber\\
&=& m_{[-1]}S(1_{(2')})\ot 1_{(1')}m_{[0]}.{\label{eq:2.2.2}}\end{aligned}$$ Now $$\begin{aligned}
&&\hspace*{-15mm}
(h_{(1)}m)_{[-1]}h_{(2)}\ot (h_{(1)}m)_{[0]}\equal{{(\ref{eq:2.2.1})}}
h_{(1)}m_{[-1]}S(h_{(3)})h_{(4)}\ot h_{(2)}m_{[0]}\\
&=& h_{(1)}m_{[-1]}\varepsilon_s(h_{(3)})\ot h_{(2)}m_{[0]}
\equal{{(\ref{eq:1.6.3})}}h_{(1)}m_{[-1]}S(1_{(2)})\ot h_{(2)}1_{(1)}m_{[0]}\\
&\equal{{(\ref{eq:2.2.2})}}&
h_{(1)}m_{[-1]}\ot h_{(2)}m_{[0]},\end{aligned}$$ as needed.

[\[co:2.2bis\]]{} Let $M$ be a left-left Yetter-Drinfeld module. For all $y\in H_s$, $z\in H_t$ and $m\in M$, we have $${\label{eq:2.2bis.1}}
\lambda(zm)=zm_{[-1]}\ot m_{[0]}~~;~~\lambda(ym)=m_{[-1]}S(y)\ot m_{[0]}.$$

$$\begin{aligned}
\lambda(zm)&\equal{(\ref{eq:1.1.4},\ref{eq:2.2.1})}&
1_{(1)}zm_{[-1]}S(1_{(3)})\ot 1_{(2)}m_{[0]}\\
&\equal{{(\ref{eq:1.1.6})}}&
z1_{(1)}m_{[-1]}S(1_{(3)})\ot 1_{(2)}m_{[0]}
\equal{{(\ref{eq:2.2.1})}} zm_{[-1]}\ot m_{[0]}.\end{aligned}$$

The other assertion is proved in a similar way.

[\[co:2.3\]]{} Let $M$ be a left-left Yetter-Drinfeld module over a weak Hopf algebra with bijective antipode. Then we have the following identities, for all $m\in M$: $${\label{eq:2.3.1}}
1_{(1)}m_{[0]}\ot 1_{(2)}S^{-1}(m_{[-1]})= m_{[0]}\ot S^{-1}(m_{[-1]});$$ $${\label{eq:2.3.2}}
\varepsilon_s(S^{-2}(m_{[-1]}))m_{[0]}=m.$$

Apply $S^{-1}$ to the first factor of [(\[eq:2.2.2\])]{}, and then switch the two tensor factors. Then we obtain [(\[eq:2.3.1\])]{}. [(\[eq:2.3.2\])]{} is proved as follows: $$\begin{aligned}
&&\hspace*{-2cm}
m=\varepsilon_t(m_{[-1]})m_{[0]}\equal{{(\ref{eq:2.1.3})}}
\varepsilon(m_{[-1]})m_{[0]}
= \varepsilon(S^{-1}(m_{[-1]}))m_{[0]}\\
&\equal{{(\ref{eq:2.3.1})}}&
\varepsilon(1_{(2)}S^{-1}(m_{[-1]}))1_{(1)}m_{[0]}
\equal{{(\ref{eq:1.6.2})}}\varepsilon_s(S^{-2}(m_{[-1]}))m_{[0]}.\end{aligned}$$

The category of left-left Yetter-Drinfeld modules and left $H$-linear, left $H$-colinear maps will be denoted by ${}_H^H{\mathcal{YD}}$.

[\[ex:2.3\]]{} Let $G$ be a groupoid, and $kG$ the corresponding groupoid algebra. Then $kG$ is a weak Hopf algebra. Let $M$ be a left-left Yetter-Drinfeld module. Then $M$ is a $kG$-comodule, so $M$ is graded by the set $G$, that is $$M=\bigoplus_{\sigma\in G_1} M_\sigma,$$ and $\lambda(m)=\sigma\ot m$ if and only if $m\in M_\sigma$, or ${\rm deg}(m)=\sigma$.\
Recall that the unit element of $kG$ is $1=\sum_{x\in G_0} x$, where $x$ is the identity morphism of the object $x\in G_0$. Take $m\in M_\sigma$. Using [(\[eq:2.2.1\])]{}, we find $$\lambda(m)=\lambda(1m)=\sum_{x\in G} x\sigma x\ot xm=0,$$ unless $s(\sigma)=\tau(\sigma)=x$. So we have $$M= \bigoplus_{\sigma\in G_1\atop s(\sigma)=t(\sigma)} M_\sigma.$$ Take $m\in M_\sigma$, with $s(\sigma)=\tau(\sigma)$, and $\tau\in G_1$. It follows from [(\[eq:2.2.1\])]{} that $\lambda(\tau m)= \tau\sigma\tau^{-1}\ot \tau m =0$, unless $s(\tau)=x$. If $s(\tau)=x$, then ${\rm deg}(\tau m)=\tau\sigma\tau^{-1}$.

[\[th:2.4\]]{} Let $H$ be a weak bialgebra. Then the category ${}_H^H{\mathcal{YD}}$ is isomorphic to the weak left center ${\mathcal{W}}_l({}_H{\mathcal{M}})$ of the category of left $H$-modules. If $H$ is a weak Hopf algebra with bijective antipode, then ${}_H^H{\mathcal{YD}}$ is isomorphic to the left center ${\mathcal{Z}}_l({}_H{\mathcal{M}})$

We will restrict to a brief description of the connecting functors; for more detail (in the left-right case), we refer to [@Nenciu Lemma 4.3]. Take $(M,\sigma_{M,-})\in {\mathcal{W}}_l({}_H{\mathcal{M}})$. For each left $H$-module $V$, we have a map $\sigma_{M,V}:\ M\ot_t V\to V\ot_t M$ in ${}_H{\mathcal{M}}$. We will show that the map $$\lambda:\ M\to H\ot_t M,~~\lambda(m)=\sigma_{M,H}(1_{(1)}m\ot 1_{(2)})=
m_{[-1]}\ot m_{[0]}$$ makes $M$ into a Yetter-Drinfeld module. Conversely, let $(M,\lambda)$ is a Yetter-Drinfeld module; a natural transformation $\sigma$ is then defined by the formula $${\label{eq:2.4.1}}
\sigma_{M,V}(1_{(1)}m\ot 1_{(2)}v)=m_{[-1]}v\ot m_{[0]}.$$ Straightforward computations show that $(M,\sigma)\in {\mathcal{W}}_l({}_H{\mathcal{M}})$. If $H$ is a Hopf algebra with invertible antipode, then the inverse of $\sigma_{M,V}$ is $${\label{eq:2.4.2}}
\sigma_{M,V}^{-1}(1_{(1)}v\ot 1_{(2)}m)=m_{[0]}\ot S^{-1}(m_{[-1]})v.$$

From now on, we assume that $H$ is a weak Hopf algebra with bijective antipode. Since the left center of a monoidal category is a braided monoidal category, it follows from [Theorem \[th:2.4\]]{} that ${}_H^H{\mathcal{YD}}$ is a braided monoidal category; a direct but long proof can be given: see [@Nenciu Prop. 2.7]. The monoidal structure can be computed using [(\[eq:center2\])]{}. Take $M,N\in {}_H^H{\mathcal{YD}}$, the $H$-coaction on $M\ot_tN$ is given by the formula $$\lambda(1_{(1)}m\ot 1_{(2)}n)=
((\sigma_{M,H}\ot N)\circ (M\ot \sigma_{N,H}))
(1_{(1')}(1_{(1)}m\ot 1_{(2)}n)\ot 1_{(2')}).$$ Observe that $$\begin{aligned}
x&=& 1_{(1')}(1_{(1)}m\ot 1_{(2)}n)\ot 1_{(2')}\\
&=& 1_{(1')}1_{(1)}m\ot 1_{(1'')}1_{(2')}1_{(2)}n\ot 1_{(2'')}=
1_{(1)}m\ot 1_{(1'')}1_{(2)}n\ot 1_{(2'')},\end{aligned}$$ so that $$\begin{aligned}
&&\hspace*{-2cm}
(M\ot \sigma_{N,H})(x)=
1_{(1)}m\ot (1_{(2)}n)_{[-1]}\ot (1_{(2)}n)_{[0]}\\
&=& 1_{(1)}m\ot 1_{(2)}n_{[-1]}S(1_{(4)})\ot 1_{(3)}n_{[0]}\\
&=& 1_{(1)}m\ot 1_{(2)}1_{(1')}n_{[-1]}S(1_{(3')})\ot 1_{(2')}n_{[0]}\\
&=& 1_{(1)}m\ot 1_{(2)}n_{[-1]}\ot n_{[0]}\end{aligned}$$ and $${\label{eq:2.5.1}}
\lambda(1_{(1)}m\ot 1_{(2)}n)=m_{[-1]}n_{[-1]}\ot m_{[0]}\ot n_{[0]}.$$ We compute the left $H$-coaction on $H_t$ using [(\[eq:center2bis\])]{} and [(\[eq:2.4.1\])]{}. For any $z\in H_t$, this gives $$\begin{aligned}
\lambda(z)&=& \sigma_{H_t,H}((1_{(1)}\leftact z)\ot 1_{(2)})=
 r_M^{-1}(l_M((1_{(1)}\leftact z)\ot 1_{(2)}))\nonumber\\
&=&
r_M^{-1}(z)= 1_{(1)}z \ot 1_{(2)}=\Delta(z).{\label{eq:2.5.2}}\end{aligned}$$ The braiding and its inverse are given by the formulas $$\sigma_{M,N}(1_{(1)} m\ot 1_{(2)}n)=m_{[-1]}n\ot m_{[0]}~~;~~
\sigma^{-1}_{M,N}(1_{(1)} n\ot 1_{(2)}m)=
m_{[0]}\ot S^{-1}(m_{[-1]})n.$$

A left-right Yetter-Drinfeld module is a $k$-module with a left $H$-action and a right $H$-coaction such that the following conditions hold, for all $m\in M$ and $h\in H$: $$\begin{aligned}
&&\hspace*{-2cm}\rho(m)= m_{[0]}\ot m_{[1]}\in M\ot_t H;{\label{eq:2.6.1}}\\
&&\hspace*{-2cm} h_{(1)}m_{[0]}\ot h_{(2)}m_{[1]}=(h_{(2)}m)_{[0]}\ot 
(h_{(2)}m)_{[1]}h_{(1)}.{\label{eq:2.6.2}}\end{aligned}$$ The category of left-right Yetter-Drinfeld modules and left $H$-linear right $H$-colinear maps is denoted by ${}_H{\mathcal{YD}}^H$.

[\[pr:2.6\]]{} Let $H$ be a weak Hopf algebra with bijective antipode. Then the category ${}_H{\mathcal{YD}}^H$ is isomorphic to the right center ${\mathcal{Z}}_r({}_H{\mathcal{M}})$.

Take $(M, \tau_{-,M})\in {\mathcal{Z}}_r({}_H{\mathcal{M}})$. We know from [Proposition \[pr:1.0.1\]]{} that $(M, \sigma_{M,-}=\tau^{-1}_{-,M})\in {\mathcal{Z}}_l({}_H{\mathcal{M}})$. Take the corresponding left-left Yetter-Drinfeld $(M,\lambda)$, as in [Theorem \[th:2.4\]]{}, and define $\rho:\ M\to M\ot H$ by $${\label{eq:2.6.0}}
\rho(m)=m_{[0]}\ot m_{[1]}=m_{[0]}\ot S^{-1}(m_{[-1]}).$$ It follows from [(\[eq:2.3.1\])]{} that $\rho(m)\in M\ot_t H$. The coassociativity of $\rho$ follows immediately from the coassociativity of $\lambda$ and the anti-comultiplicativity of $S^{-1}$. Also $$\varepsilon(m_{[1]})m_{[0]}=\varepsilon(S^{-1}(m_{[-1]}))m_{[0]}
=\varepsilon(m_{[-1]})m_{[0]}=m.$$ From [(\[eq:2.4.2\])]{}, it follows that $${\label{eq:2.6.3}}
\tau_{V,M}(1_{(1)}v\ot 1_{(2)}m)=m_{[0]}\ot m_{[1]}v.$$ In particular, $\tau_{M,H}(1_{(1)}\ot 1_{(2)}m)=\rho(m)$, and the fact that $\tau_{M,H}$ is left $H$-linear implies [(\[eq:2.6.2\])]{}. Hence $(M,\rho)$ is a left-right Yetter-Drinfeld module.\
Conversely, if $(M,\rho)$ is a left-right Yetter-Drinfeld module, then $(M,\tau_{-,M})$, with $\tau$ defined by [(\[eq:2.6.3\])]{} is an object of ${\mathcal{Z}}_r({}_H{\mathcal{M}})$.

[\[co:2.7\]]{} Let $M$ be a $k$-module with a left $H$-action and a right $H$-coaction. Then $M$ is a left-right Yetter-Drinfeld module if and only if $${\label{eq:2.7.1}}
\rho(hm)=h_{(2)}m_{[0]}\ot h_{(3)}m_{[1]}S^{-1}(h_{(1)}).$$

[\[co:2.7a\]]{} Let $M$ be a left-right Yetter-Drinfeld module. For all $y\in H_s$, $z\in H_t$ and $m\in M$, we have that $${\label{eq:2.7a.1}}
\rho(ym)=m_{[0]}\ot ym_{[1]}~~;~~\rho(zm)=m_{[0]}\ot m_{[1]}S^{-1}(z).$$

[\[co:2.7b\]]{} Let $M$ be a left-right Yetter-Drinfeld module. Then $${\label{eq:2.7b.1}}
1_{(2)}m_{[0]}\ot m_{[1]}S^{-1}(1_{(1)})=\rho(m),$$ for all $m\in M$.

Apply $S^{-1}\ot M$ to $\lambda(m)=1_{(1)}S(m_{[1]})\ot 1_{(2)}m_{[0]}$.

[\[co:2.8\]]{} The category ${}_H{\mathcal{YD}}^H$ is a braided monoidal category, isomorphic to ${}_H^H{\mathcal{YD}}^{\rm in}$.

In a similar way, we can introduce right-right and right-left Yetter-Drinfeld modules. The categories ${\mathcal{YD}}_H^H$ and ${}^H{\mathcal{YD}}_H$ of right-right and right-left Yetter-Drinfeld modules are isomorphic to the right and left center of ${\mathcal{M}}_H$. Let us summarize the results.\
A right-right Yetter-Drinfeld module is a $k$-module $M$ with a right $H$-action and a right $H$-coaction such that $$\begin{aligned}
&&\hspace*{-2cm}\rho(m)= m_{[0]}\ot m_{[1]}\in M\ot_s H;{\label{eq:2.9.1}}\\
&&\hspace*{-2cm} m_{[0]}h_{(1)}\ot m_{[1]}h_{(2)}=(mh_{(2)})_{[0]}\ot 
h_{(1)}(mh_{(2)})_{[1]};{\label{eq:2.9.2}}\end{aligned}$$ or, equivalently, $${\label{eq:2.9.3}}
\rho(mh)=m_{[0]}h_{(2)}\ot S(h_{(1)})m_{[1]}h_{(3)}.$$ The counit condition $m=\varepsilon(m_{[1]})m_{[0]}$ is equivalent to $$m= m_{[0]}\varepsilon(m_{[1]}).$$ The natural isomorphism $\tau_{-,M}$ corresponding to $(M,\rho)\in {\mathcal{YD}}_H^H$ and its inverse are given by the formulas $${\label{eq:2.9.4}}
\tau_{M,V}(v1_{(1)}\ot m1_{(2)})=m_{[0]}\ot mv_{[1]}~~;~~
\tau^{-1}_{M,V}(m1_{(1)}\ot v1_{(2)})=vS^{-1}(m_{[1]})\ot m_{[0]}.$$ Furthermore $$m_{[0]}\varepsilon_t(S^{-2}(m_{[1]}))=m,$$ and $S^{-1}(m_{[1]})\ot m_{[0]}\in H\ot_s M$.\
The monoidal structure on ${\mathcal{YD}}_H^H$ is given by the formula $$\rho(m1_{(1)}\ot n1_{(2)})=m_{[0]}\ot n_{[0]}\ot m_{[1]}n_{[1]}.$$ The braiding is given by [(\[eq:2.9.4\])]{}. The category ${\mathcal{YD}}_H^H$ is isomorphic as a braided monoidal category to ${\mathcal{Z}}_r({\mathcal{M}}_H)$.\
Let $M$ be a right $H$-module and a left $H$-comodule. $M$ is a right-left Yetter-Drinfeld module if one of the three following equivalent conditions is satisfied, for all $m\in M$ and $h\in H$:\
1) $\lambda(m)\in H\ot_sM$ and $$h_{(2)}(mh_{(1)})_{[0]}\ot (mh_{(1)})_{[1]}=
m_{[-1]}h_{(1)}\ot m_{[0]}h_{(2)},$$ 2) $\lambda(mh)=S^{-1}(h_{(3)})m_{[-1]}h_{(1)}\ot m_{[0]}h_{(2)}$;\
3) $(M,\rho)$, with $\rho(m)=m_{[0]}\ot S(m_{[-1]})$ is a right-right Yetter-Drinfeld module.\
The category of right-left Yetter-Drinfeld modules, ${}^H{\mathcal{YD}}_H$, is a braided monoidal category. The monoidal structure and the braiding are given by $$\lambda(m1_{(1)}\ot n1_{(2)})=m_{[-1]}n_{[-1]}\ot m_{[0]}\ot n_{[0]};$$ $$\sigma_{M,N}(m1_{(1)}\ot n1_{(2)})=nm_{[-1]}\ot m_{[0]}.$$ As a braided monoidal category, ${}^H{\mathcal{YD}}_H$ is isomorphic to ${\mathcal{Z}}_l({\mathcal{M}}_H)$ and $({\mathcal{YD}}_H^H)^{\rm in}$.\
The antipode $S:\ H\to H^{\rm op,cop}$ is an isomorphism of weak Hopf algebras. Observe that the target map of $H^{\rm op,cop}$ is $\varepsilon_s$, and that its source map is $\varepsilon_t$. Thus $S$ induces an isomorphism between the monoidal categories ${}_H{\mathcal{M}}$ and ${}_{H^{\rm op,cop}}{\mathcal{M}}$. We also have a monoidal isomorphism $F:\ {}_{H^{\rm op,cop}}{\mathcal{M}}\to \ol{{\mathcal{M}}}_H$, given by $$F(M)=M,~~mh=h^{\rm op,cop}m.$$ indeed, in ${}_{H^{\rm op,cop}}{\mathcal{M}}$, $M\ot_t N$ is generated by elements of the form $1_{(2)}m\ot 1_{(1)}n$, and $F(M\ot_t N)$ is generated by elements of the form $m1_{(2)}\ot n1_{(1)}$. $F(N)\ot_sF(M)$ is generated by elements of the form $n1_{(1)}\ot m1_{(2)}$, and it follows that the switch map is an isomorphism $F(M\ot_t N)\to F(N)\ot_s F(M)$. We conclude from [Proposition \[pr:1.0.2\]]{} that we have isomorphisms of braided monoidal categories $${}_H^H{\mathcal{YD}}\cong {\mathcal{Z}}_l({}_H{\mathcal{M}})\cong {\mathcal{Z}}_l({}_{H^{\rm op,cop}}{\mathcal{M}})
\cong {\mathcal{Z}}_l(\ol{{\mathcal{M}}}_H)\cong \ol{{\mathcal{Z}}_r({\mathcal{M}}_H)}\cong \ol{{\mathcal{YD}}}_H^H.$$ This isomorphism can be described explicitely as follows: $$F:\ {}_H^H{\mathcal{YD}}\to \ol{{\mathcal{YD}}}_H^H,~~F(M)=M,$$ with $$m\cdot h=S^{-1}(h)m~~;~~\rho(m)=m_{[0]}\ot S(m_{[-1]}).$$ We summarize our results as follows:

[\[th:2.10\]]{} Let $H$ be a weak Hopf algebra with bijective antipode. Then we have the following isomorphisms of braided monoidal categories: $${}_H^H{\mathcal{YD}}\cong {{}_H{\mathcal{YD}}^H}^{\rm in}\cong \ol{{\mathcal{YD}}_H^H}
\cong \ol{{}^H{\mathcal{YD}}_H}^{\rm in}.$$

Yetter-Drinfeld modules are Doi-Hopf modules
============================================

[\[se:3\]]{} It was shown in [@cmz] that Yetter-Drinfeld modules (over a classical Hopf algebra) can be considered as Doi-Hopf modules, and, a fortiori, as entwined modules, and as comodules over a coring (see [@BW]). Weak Doi-Hopf modules were introduced by Böhm [@Bohm], and they are special cases of weak entwined modules (see [@CDG]), and these are in turn examples of comodules over a coring (see [@BW]). In this Section, we will show that Yetter-Drinfeld modules over weak Hopf algebras are special cases of weak Doi-Hopf modules. We will discuss the left-right case.

[\[pr:3.1\]]{} Let $H$ be a weak Hopf algebra with a bijective antipode. Then $H$ is a right $H\ot H^{\rm op}$-comodule algebra, with $H$-coaction $$\rho(h)=h_{(2)}\ot S^{-1}(h_{(1)})\ot h_{(3)}.$$

It is easy to verify that $H$ is a right $H\ot H^{\rm op}$-comodule and that $\rho(hk)=\rho(h)\rho(k)$. Recall that $H_t={{\rm Im}\,}(\varepsilon_t)={{\rm Im}\,}(\ol{\varepsilon}_t)$. The target map of $H^{\rm op}\ot H$ is $\ol{\varepsilon}_t\ot\varepsilon_t$. We now have $$\begin{aligned}
&&\hspace*{-2cm}
1_{[0]}\ot (\ol{\varepsilon}_t\ot\varepsilon_t)(1_{[1]})=
1_{(2)}1_{(1')}\ot \ol{\varepsilon}_t(S^{-1}(1_{(1)}))\ot 
\varepsilon_t(1_{(2')})\\
&=& 1_{(2)}1_{(1')}\ot S^{-1}(1_{(1)})\ot 1_{(2')}=\rho(1),\end{aligned}$$ where we used the fact that $S^{-1}(1_{(1)})\ot 1_{(2)}\in H_t\ot H_t$.

[\[pr:3.2\]]{} Let $H$ be a weak Hopf algebra with a bijective antipode. Then $H$ is a left $H^{\rm op}\ot H$-module coalgebra with left action $$(k\ot h)\triangleright c= hck.$$

We easily compute that $$\begin{aligned}
&&\hspace*{-2cm}
\varepsilon((m\ot l)(k_{(2)}\ot h_{(2)}))\varepsilon((k_{(1)}\ot h_{(1)})\triangleright c)\\
&=& \varepsilon(k_{(2)}m)\varepsilon(lh_{(2)})\varepsilon(h_{(1)}ck_{(1)})\\
&=& \varepsilon(lhckm)=\varepsilon(((m\ot l)(k\ot h))\triangleright c).\end{aligned}$$ The other conditions are easily verified.

[\[co:3.3\]]{} Let $H$ be a weak Hopf algebra with bijective antipode. Then we have a weak Doi-Hopf datum $(H^{\rm op}\ot H, H,H)$ and the categories ${}_H{\mathcal{M}}(H^{\rm op}\ot H)^H$ and ${}_H{\mathcal{YD}}^H$ are isomorphic.

The compatibility relation [(\[eq:3.1.2\])]{} reduces to [(\[eq:2.7.1\])]{}.

As we have seen in [Section \[se:1.4\]]{}, weak Doi-Hopf modules are special cases of entwined modules. The entwining map $\psi:\ H\ot H\to H\ot H$ corresponding to the weak Doi-Hopf datum $(H^{\rm op}\ot H, H,H)$ is given by $${\label{eq:3.3.1}}
\psi(h\ot k)=h_{(2)}\ot h_{(3)}kS^{-1}(h_{(1)}).$$

The Drinfeld double
===================

[\[se:4\]]{} Now we consider the particular case where $H$ is finitely generated and projective as a $k$-module, with finite dual basis $\{(h_i,h_i^*)~|~i=1,\cdots,n\}$. Then $H^*$ is also a weak Hopf algebra, in view of the selfduality of the axioms of a weak Hopf algebra. Recall that the comultiplication is given by the formula $\lan \Delta(h^*),h\ot k\ran =\lan h^*,hk\ran$; the counit is evaluation at $1$. Also recall that $H^*$ is an $H$-bimodule, with left and right $H$-action $$\lan h\leftact h^*\rightact k, l\ran =\lan h^*, klh\ran,$$ or $${\label{eq:4.0}}
h\leftact h^*\rightact k=\lan h_{(1)}^*,k\ran \lan h_{(3)}^*,h\ran h_{(2)}^*.$$ Using [(\[eq:1.41\])]{}, we find a weak smash product structure $(H,H^*,R)$, with $R:\ H^*\ot H\to H\ot H^*$ given by $$\begin{aligned}
R(h^*\ot h)&=&
\hbox{$\sum_i$} \lan h^*, h_{(3)}h_iS^{-1}(h_{(1)})\ran h_{(2)}\ot h_i^*\nonumber\\
&=& \hbox{$\sum_i$} \lan S^{-1}(h_{(1)})\leftact h^*\rightact h_{(3)}, h_i\ran h_{(2)}\ot h_i^*\nonumber\\
&=& h_{(2)} \ot \Bigl(S^{-1}(h_{(1)})\leftact h^*\rightact h_{(3)}\Bigr).
{\label{eq:4.1}}\end{aligned}$$

From [Section \[se:1.4\]]{}, we know that $H\#_R H^*$, which we will also denote by $H\bowtie H^*$, is an associative algebra with preunit $1\# \varepsilon$. Using [(\[eq:smash\])]{}, we compute the multiplication rule on $H\bowtie H^*$. $$\begin{aligned}
&&\hspace*{-2cm} (h\bowtie h^*)(k\bowtie k^*)=
\hbox{$\sum_i$} \lan h^*, k_{(3)}h_iS^{-1}(k_{(1)})\ran hk_{(2)}
\bowtie h_i^**k^*\nonumber\\
&=& hk_{(2)}\bowtie (S^{-1}(k_{(1)})\leftact h^*\rightact k_{(3)})*k^*
{\label{eq:4.2}}\\
&=& hk_{(2)} \bowtie \lan h_{(1)}^*, k_{(3)}\ran \lan h_{(3)}^*,S^{-1}(k_{(1)})\ran
h_{(2)}^**k^*{\label{eq:4.3}}.\end{aligned}$$ We have a projection $p:\ H\bowtie H^*\to H\bowtie H^*$, $$p(h\bowtie h^*)=(1\bowtie \varepsilon)(h\bowtie h^*)=(h\bowtie h^*)(1\bowtie \varepsilon)= (h\bowtie h^*)(1\bowtie \varepsilon)^2,$$ and $D(H)=\ol{H\bowtie H^*}= (H\bowtie H^*)/{{\rm Ker}\,}p$ is a $k$-algebra with unit $[1\bowtie \varepsilon]$, which we call the [*Drinfeld double*]{} of $H$. $D(H)$ is also isomorphic to $\ul{H\bowtie H^*}={{\rm Im}\,}(p)$, which is a $k$-algebra with unit $(1\bowtie \varepsilon)^2$. Observe that the multiplication rule [(\[eq:4.3\])]{} is the same as in [@Bohm; @Nenciu]. We show that the ideal $J$ that is divided out in [@Bohm; @Nenciu] is equal to ${{\rm Ker}\,}p$, and this will imply that $D(H)$ is equal to the Drinfeld double introduced in [@Bohm; @Nenciu]. We first need some Lemmas.

[\[le:4.1\]]{} Let $H$ a weak bialgebra. For all $h^*\in H^*$, $y\in H_s$ and $z\in H_t$, we have $$\begin{aligned}
h^**(y\leftact\varepsilon)&=&
\lan h_{(2)}^*, y\ran  h_{(1)}^*=y\leftact h^*{\label{eq:4.1.1}}\\
h^**(\varepsilon\rightact y)&=&
\lan h_{(1)}^*, y\ran  h_{(2)}^*=h^*\rightact y{\label{eq:4.1.2}}\\
(z\leftact \varepsilon)*h^*&=&
\lan h_{(2)}^*,z\ran h_{(1)}^*=z\leftact h^*{\label{eq:4.1.3}}\\
(\varepsilon\rightact z)*h^*&=&
\lan h_{(1)}^*,z\ran h_{(2)}^*=h^*\rightact z{\label{eq:4.1.4}}\end{aligned}$$

We only prove [(\[eq:4.1.3\])]{}. For all $h\in H$, we have $$\begin{aligned}
&&\hspace*{-2cm}
\lan (z\leftact \varepsilon)*h^*, h\ran=
\lan \varepsilon, h_{(1)}z\ran \lan h^*,h_{(2)}\ran
= \lan \varepsilon, h_{(1)}1_{(1)}z\ran \lan h^*,h_{(2)}1_{(2)}\ran\\
&=& \lan \varepsilon*h^*, hz\ran
= \lan h^*, hz\ran=\lan z\leftact h^*,h\ran = \lan h_{(2)}^*,z\ran
\lan h_{(1)}^*,h\ran.\end{aligned}$$

[\[le:4.1a\]]{} Let $H$ be a weak Hopf algebra with bijective antipode. For all $y\in H_s$, $z\in H_t$, we have $${\label{eq:4.1.5}}
S^{-1}(z)\leftact \varepsilon=z\leftact\varepsilon~~{\rm and}~~
\varepsilon\rightact y=\varepsilon\rightact S^{-1}(y).$$

For all $h\in H$, we have $$\begin{aligned}
&&\hspace*{-2cm}
\lan S^{-1}(z)\leftact \varepsilon,h\ran= \varepsilon(hS^{-1}(z))\equal{{(\ref{eq:1.1.2})}}
\varepsilon(h1_{(1)})\varepsilon(1_{(2)}S^{-1}(z))
\equal{\ref{le:1.5}}
\varepsilon(h1_{(1)})\varepsilon(zS(1_{(2)}))\\
&\equal{{(\ref{eq:1.6.3})}}&
\varepsilon(h_{(1)})\varepsilon(z\varepsilon_s(h_{(2)}))
\equal{{(\ref{eq:1.1.6})}}\varepsilon(\varepsilon_s(h)z)
\equal{{(\ref{eq:1.1.3a})}} \varepsilon(hz)=\lan z\leftact\varepsilon, h\ran.\end{aligned}$$ The second statement can be proved in a similar way.

[\[pr:4.2\]]{} Let $H$ be a finitely generated projective weak Hopf algebra. Then ${{\rm Ker}\,}(p)$ is the $k$-linear span $J$ of elements of the form $$A=hz\bowtie h^*-h\bowtie (z\leftact \varepsilon)*h^*~~{\rm and}~~
B=hy\bowtie h^*-h\bowtie (\varepsilon\rightact y)*h^*,$$ where $h\in H$, $h^*\in H^*$, $y\in H_s$ and $z\in H_t$.

$A\in {{\rm Ker}\,}(p)$ since $$\begin{aligned}
&&\hspace*{-2cm}
(1\bowtie \varepsilon)(hz\bowtie h^*)\equal{{(\ref{eq:4.3})}}
h_{(2)}1_{(2)}\bowtie \varepsilon_{(2)}*h^*
\lan \varepsilon_{(1)},h_{(3)}1_{(3)}\ran
\lan \varepsilon_{(3)},S^{-1}(h_{(1)}1_{(1)}z)\ran\\
&=& h_{(2)}\bowtie \varepsilon_{(2)}*h^*
\lan \varepsilon_{(1)},h_{(3)}\ran
\lan \varepsilon_{(3)},S^{-1}(z)\ran
\lan \varepsilon_{(4)},S^{-1}(h_{(1)})\ran\\
&\equal{{(\ref{eq:4.1.1})}}& h_{(2)}\bowtie (\varepsilon_{(2)}*
(S^{-1}(z)\leftact \varepsilon)*h^*)
\lan \varepsilon_{(1)},h_{(3)}\ran
\lan \varepsilon_{(3)},S^{-1}(h_{(1)})\ran\\
&\equal{{(\ref{eq:4.3})}}& (1\bowtie \varepsilon)
\bigl(h\bowtie ((S^{-1}(z)\leftact \varepsilon)*h^*)\bigr)
\equal{{(\ref{eq:4.1.5})}} (1\bowtie \varepsilon)(h\bowtie (z\leftact\varepsilon)*h^*).\end{aligned}$$ In a similar way, $B\in {{\rm Ker}\,}(p)$: $$\begin{aligned}
&&\hspace*{-2cm}
(1\bowtie \varepsilon)(hy\bowtie h^*)\equal{{(\ref{eq:4.3})}}
(h_{(2)}1_{(2)}\bowtie \varepsilon_{(2)}^**h^*)
\lan \varepsilon_{(1)},h_{(3)}1_{(3)}y\ran
\lan\varepsilon_{(3)}, S^{-1}(h_{(1)}1_{(1)})\ran\\
&=& (h_{(2)}\bowtie \varepsilon_{(3)}^**h^*)
\lan \varepsilon_{(1)},h_{(3)}\ran
\lan \varepsilon_{(2)},y\ran
\lan\varepsilon_{(4)}, S^{-1}(h_{(1)})\ran\\
&\equal{{(\ref{eq:4.1.2})}}&
(h_{(2)}\bowtie (\varepsilon_{(2)}\rightact y)*h^*)
\lan \varepsilon_{(1)},h_{(3)}\ran\lan\varepsilon_{(3)}, S^{-1}(h_{(1)})\ran\\
&\equal{{(\ref{eq:4.3})}}& (1\bowtie \varepsilon)
(h\bowtie (\varepsilon\rightact y)*h^*).\end{aligned}$$ This shows that $J\subset {{\rm Ker}\,}(p)$. We now compute for all $h\in H$ and $h^*\in H^*$ that $$(h\bowtie h^*)(1\bowtie\varepsilon)\equal{{(\ref{eq:4.3})}}
(h1_{(2)}1_{(1')}\bowtie h_{(2)}^*)
\lan h_{(1)}^*,1_{(2')}\ran \lan h_{(3)}^*, S^{-1}(1_{(1)})\ran,$$ and $$\begin{aligned}
&&\hspace*{-20mm}
\Bigl(h\bowtie (S^{-1}(1_{(2)})\leftact\varepsilon)*(\varepsilon\rightact 1_{(1')})
*h_{(2)}^*\Bigr) \lan h_{(1)}^*,1_{(2')}\ran \lan h_{(3)}^*,S^{-1}(1_{(1)})\ran\\
&\equal{{(\ref{eq:4.0})}}&
\Bigl(h\bowtie \varepsilon_{(1)}*\varepsilon_{(2')}*h_{(2)}^*\Bigr)
\lan \varepsilon_{(2)}, S^{-1}(1_{(2)})\ran \lan \varepsilon_{(1')}, 1_{(1')}\ran\\
&&\hspace*{1cm}
\lan h_{(1)}^*,1_{(2')}\ran \lan h_{(3)}^*,S^{-1}(1_{(1)})\ran\\
&=& \Bigl(h\bowtie \varepsilon_{(1)}*\varepsilon_{(2')}*h_{(2)}^*\Bigr)
\lan \varepsilon_{(1')}*h_{(1)}^*, 1\ran
\lan \varepsilon_{(2)}*h_{(3)}^*, S^{-1}(1)\ran\\
&=& \Bigl(h\bowtie \varepsilon_{(1)}*h_{(1)}^*\Bigr)
\lan \varepsilon_{(2)}*h_{(2)}^*, 1\ran=
 h\bowtie (\varepsilon* h^*)=h\bowtie h^*.\end{aligned}$$ Observing that $$\begin{aligned}
&&\hspace*{-2cm}
hzy\bowtie h^* - 
h\bowtie ((S^{-1}(z)\leftact \varepsilon)*(\varepsilon\rightact y)*h^*)\\
&=& hzy\bowtie h^* - hz\bowtie (\varepsilon\rightact y) *h^*)\\
&+&hz\bowtie (\varepsilon\rightact y) *h^*)
- h\bowtie ((S^{-1}(z)\leftact \varepsilon)*(\varepsilon\rightact y)*h^*)\in J,\end{aligned}$$ it follows that $(h\bowtie h^*)(1\bowtie\varepsilon)- (h\bowtie h^*)\in J$, for all $h\in H$ and $h^*\in H^*$. If $x\in {{\rm Ker}\,}(p)$, then $x(1\bowtie \varepsilon)=0$, and $x= x-x(1\bowtie \varepsilon)\in J$. We conclude that ${{\rm Ker}\,}(p)\subset J$, finishing our proof.

We now recall the following results from [@NVT]. On $H^*\ot H$, there exists an associative multiplication $$\begin{aligned}
(h^*\ot h)(k^*\ot k)&=&
k_{(2)}^*h^*\ot h_{(2)}k \lan S(h_{(1)}),k_{(1)}^*\ran
\lan h_{(3)}, k_{(3)}^*\ran\\
&=& (h_{(3)}\leftact k^*\rightact S(h_{(1)}))*h^*\ot h_{(2)}k.\end{aligned}$$ The $k$-module $I$ generated by elements of the form $$A'=h^*\ot hz-(\varepsilon\rightact z)h^*\ot h~~{\rm and}~~
B'=h^*\ot yh-(y\leftact\varepsilon)h^*\ot h$$ is a two-sided ideal of $H^*\ot H$. The quotient $D'(H)=(H^*\ot H)/I$ is an algebra with unit element $\varepsilon\ot 1$. It is a weak Hopf algebra, with the following comultiplication, counit and antipode: $$\begin{aligned}
&&\hspace*{-2cm} \Delta[h^*\ot h]=[h^*_{(1)}\ot h_{(1)}]\ot [h^*_{(2)}\ot h_{(2)}]
{\label{eq:4.3.1}}\\
&&\hspace*{-2cm} \varepsilon[h^*\ot h]=\lan h^*,\varepsilon_t(h)\ran{\label{eq:4.3.2}}\\
&&\hspace*{-2cm} S[h^*\ot h]=[S^{-1}(h_{(2)}^*)\ot S(h_{(2)})]
\lan h_{(1)}^*,h_{(1)}\ran \lan h_{(3)}^*,S(h_{(3)})\ran {\label{eq:4.3.3}}\end{aligned}$$

[\[pr:4.3\]]{} The $k$-linear isomorphism $$f:\ H\bowtie H^*\to H^*\ot H,~~f(h\bowtie h^*)=h^*\ot S^{-1}(h)$$ is anti-multiplicative, and induces an algebra isomorphism $f:\ D(H)\to D'(H)^{\rm op}$.

Let us first prove that $f$ reverses the multiplication. Indeed, $$\begin{aligned}
&&\hspace*{-2cm}
f(k\bowtie k^*)f(h\bowtie h^*)= (k^*\ot S^{-1}(k))(h^*\ot S^{-1}(h))\\
&=& (S^{-1}(k_{(1)})\leftact h^*\rightact k_{(3)})*k^*\ot S^{-1}(k_{(2)})S^{-1}(h)\\
&=& f((h\bowtie h^*)(k\bowtie k^*)).\end{aligned}$$ Using [Lemma \[le:4.1a\]]{}, we easily compute that $f(J)=I$, and the result follows.

Let us now define a comultiplication, counit and antipode on $D(H)$, in such a way that $f:\ D(H)\to D'(H)$ is an isomorphism of Hopf algebras. Obviously, the comultiplication is given by the formula $${\label{eq:4.4.1}}
\Delta[h\bowtie h^*]=[h_{(2)}\bowtie h^*_{(1)}]\ot [h_{(1)}\bowtie h^*_{(2)}].$$ The counit is computed as follows: $$\varepsilon[h\bowtie h^*]=\varepsilon[h^*\ot S^{-1}(h)]
\equal{{(\ref{eq:4.3.2})}}\lan h^*,\varepsilon_t(S^{-1}(h))\ran
\equal{{(\ref{eq:1.4.1})}}\lan h^*,1_{(2)}\ran \lan \varepsilon,h1_{(1)}\ran.
{\label{eq:4.4.2}}$$ Since the antipode of $H$ is the inverse of the antipode of $H^{\rm op}$, the antipode of $D'(H)$ is transported to the inverse of the antipode of $D(H)$. We find $$\begin{aligned}
&&\hspace*{-2cm}S^{-1}[h\bowtie h^*]=
(f^{-1}\circ S\circ f)[h\bowtie h^*]=
f^{-1}(S[h^*\ot S^{-1}(h)])\nonumber\\
&=& f^{-1}[S^{-1}(h_{(2)}^*)\ot h_{(2)}]\lan h_{(1)}^*,S^{-1}(h_{(3)})\ran
\lan h_{(3)}^*,h_{(1)}\ran\nonumber\\
&=& [S(h_{(2)})\bowtie S^{-1}(h_{(2)}^*)]\lan h_{(1)}^*,S^{-1}(h_{(3)})\ran
\lan h_{(3)}^*,h_{(1)}\ran{\label{eq:4.4.3}}\end{aligned}$$ The antipode $S$ is then given by the formula $${\label{eq:4.4.4}}
S[h\bowtie h^*]=[S^{-1}(h_{(2)})\bowtie S(h_{(2)}^*)]
\lan h_{(1)}^*,S^{-1}(h_{(3)})\ran \lan h_{(3)}^*,h_{(1)}\ran$$ Indeed, $$\begin{aligned}
&&\hspace*{-15mm}
S(S^{-1}[h\bowtie h^*])\\
&=& [h_{(3)}\bowtie h_{(3)}^*]
\lan h_{(1)}^*,S^{-1}(h_{(5)})\ran \lan h_{(2)}^*,h_{(4)}\ran
\lan h_{(5)}^*,h_{(1)}\ran \lan h_{(4)}^*,S^{-1}(h_{(2)})\ran\\
&=& [h_{(3)}\bowtie h_{(2)}^*]
\lan h_{(1)}^*,S^{-1}(h_{(5)})h_{(4)}\ran
\lan h_{(2)}^*,S^{-1}(h_{(2)})h_{(1)}\ran\\
&=& [h_{(2)}\bowtie h_{(2)}^*]
\lan h_{(1)}^*,\varepsilon_t(S^{-1}(h_{(3)}))\ran
\lan h_{(3)}^*,\varepsilon_t(S^{-1}(h_{(1)}))\ran\\
&=&\varepsilon([h\bowtie h^*]_{(1)})[h\bowtie h^*]_{(2)}\varepsilon([h\bowtie h^*]_{(3)})
= [h\bowtie h^*].\end{aligned}$$ Similar arguments show that $S^{-1}(S[h\bowtie h^*])=[h\bowtie h^*]$.

[\[pr:4.4\]]{} Let $H$ be a weak Hopf algebra with bijective antipode, which is finitely generated and projective as a $k$-module. Then $D(H)$ is a weak Hopf algebra, with comultiplication, counit and antipode given by the formulas (\[eq:4.4.1\],\[eq:4.4.2\],\[eq:4.4.3\]). As a weak Hopf algebra, $D(H)$ is isomorphic to $D'(H)^{\rm op}$.

[\[pr:4.5\]]{} Let $H$ be a weak Hopf algebra with bijective antipode, which is finitely generated and projective as a $k$-module. The functor $$F:\ {}_H{\mathcal{YD}}^H\to {}_{D(H)}\ol{{\mathcal{M}}},~~F(M)=M,$$ with $$(h\bowtie h^*)m=\lan h^*,m_{[1]}\ran hm_{[0]},$$ for all $h\in H$, $h^*\in H^*$ and $m\in M$ is an isomorphism of monoidal categories.

We already know (see [(\[eq:1.42\])]{}) that $F$ is an isomorphism of categories, so we only have to show that $F$ preserves the product. Take $M,N\in {}_H{\mathcal{YD}}^H$. The right $H$-coaction on $M\ot_t N$ is given by the formula (use [(\[eq:2.5.1\])]{} and [(\[eq:2.6.0\])]{}): $$\rho(1_{(1)}m\ot 1_{(2)}n)=
m_{[0]}\ot n_{[0]}\ot n_{[1]}m_{[1]},$$ hence the left $D(H)$-action on $F(M\ot_t N)$ is the following $${\label{eq:4.5.1}}
[h\bowtie h^*](1_{(1)}m\ot 1_{(2)}n)= \lan h^*,n_{[1]}m_{[1]}\ran
h_{(1)}m_{[0]}\ot h_{(2)}n_{[0]}.$$ We now compute $$F(N)\ot_t F(M)=\{[1\bowtie \varepsilon]X~|~X\in F(N)\ot F(M)\}.$$ Observe that $$\begin{aligned}
&&\hspace*{-2cm}
[1\bowtie\varepsilon]_{(1)}n\ot [1\bowtie\varepsilon]_{(2)}m
=\lan  \varepsilon_{(1)}, n_{[1]}\ran 1_{(2)}n_{[0]}\ot 
\lan \varepsilon_{(2)}, m_{[1]}\ran 1_{(1)}m_{[0]}\\
&=&\lan\varepsilon,n_{[1]}m_{[1]}\ran 1_{(2)}n_{[0]}\ot 1_{(1)}m_{[0]}.\end{aligned}$$ We claim that the switch map $\tau:\ M\ot N\to N\ot M$ induces an isomorphism $\tau:\ F(M\ot_t N)\to F(N)\ot_t F(M)$ of $k$-modules. Indeed, take $1_{(1)}m\ot 1_{(2)}n\in M\ot_t N$. Since $M\ot_t N$ is a Yetter-Drinfeld module, we have that $\varepsilon(n_{[1]}m_{[1]})m_{[0]}\ot n_{[0]}=1_{(1)}m\ot 1_{(2)}n$, and $$\begin{aligned}
&&\hspace*{-2cm}
\tau(1_{(1)}m\ot 1_{(2)}n)
=1_{(2)}n\ot 1_{(1)}m= 1_{(2')}1_{(2)}n\ot 1_{(1')}1_{(1)}m\\
&=& \varepsilon(n_{[1]}m_{[1]})1_{(2)} n_{[0]}\ot 1_{(1)}m_{[0]}\\
&=&[1\bowtie\varepsilon]_{(1)}n\ot [1\bowtie\varepsilon]_{(2)}m
\in F(N)\ot_t F(M).\end{aligned}$$ Conversely, $$\tau([1\bowtie\varepsilon]_{(1)}n\ot [1\bowtie\varepsilon]_{(2)}m)=
\varepsilon(n_{[1]}m_{[1]})1_{(1)}m_{[0]}\ot 1_{(2)} n_{[0]}
\in F(M\ot_t N).$$ Let us now show that $\tau$ is left $D(H)$-linear. To this end, we compute the left $D(H)$-action on $F(N)\ot_t F(M)$. $$\begin{aligned}
&&\hspace*{-15mm}
[h\bowtie h^*]\tau(1_{(1)}m\ot 1_{(2)}n)=
[h\bowtie h^*](1_{(2)}n\ot 1_{(1)}m)\\
&=&[h_{(2)}\bowtie h_{(1)}^*](1_{(2)}n)\ot [h_{(1)}\bowtie h_{(2)}^*](1_{(1)}m)\\
&\equal{{(\ref{eq:2.7a.1})}}&
\lan h_{(1)}^*,n_{[1]}S^{-1}(1_{(2)})\ran h_{(2)}n_{[0]}\ot
\lan h_{(2)}^*,1_{(1)}m_{[1]}\ran h_{(1)}m_{[0]}\\
&=& \lan h^*, n_{[1]}S^{-1}(1_{(2)})1_{(1)}m_{[1]}\ran 
h_{(2)}n_{[0]}\ot h_{(1)}m_{[0]}\\
&\equal{ {(\ref{eq:4.5.1})}}& \tau\bigl([h\bowtie h^*](1_{(1)}m\ot 1_{(2)}n)\bigr)\end{aligned}$$ It also follows that $F(H_t)$ is a unit object in ${}_{D(H)}\ol{{\mathcal{M}}}$. Since the unit object in a monoidal category is unique up to automorphism, we concluce that the target space of $D(H)_t$ is isomorphic to $H_t$. This can also be seen as follows: in [@NVT], it is shown that $D'(H)_t=[\varepsilon\ot H_t]\cong H_t$. Since the target spaces of a weak Hopf algebra and its opposite coincide, it follows that $D(H)_t\cong H_t$.

Duality
=======

[\[se:5\]]{} Let $H$ be a weak Hopf algebra with bijective antipode, and ${}_H{\rm Rep}$ the category of left $H$-modules $M$ which are finitely generated projective as a $k$-module. Let $M\in {}_H{\rm Rep}$, and let $\{(n_i,n_i^*)~|~i=1,\cdots n\}$ be a finite dual basis of $M$. From [@NVT], we recall the following result. We refer to [@Kassel] for the definition of duality in a monoidal category.

[\[pr:5.1\]]{} The category ${}_H{\rm Rep}$ has left duality. The left dual of $M\in {}_H{\rm Rep}$ is $M^*={{\rm Hom}}(M,k)$ with left $H$-action defined by $${\label{eq:5.1.1}}
\lan h\cdot m^*, m\ran = \lan m^*,S(h)m\ran,$$ for all $h\in H$, $m\in M$ and $m^*\in M^*$. The evaluation map ${\rm ev}_M:\ M^*\ot_t M\to H_t$ and the coevaluation map ${\rm coev}_M:\ H_t\to M\ot_t M^*$ are defined as follows: $${\rm ev}_M(1_{(1)}\cdot m^*\ot 1_{(2)}m)=\lan m^*,1_{(1)}m\ran 1_{(2)};$$ $${\rm coev}_M(z)=z\cdot (\hbox{$\sum_i$} n_i\ot n_i^*).$$

Let $M$ be a finitely generated projective left $H$-comodule. Then $M^*$ is also a left $H$-comodule, with left $H$-coaction $\lambda:\ M^*\to
H\ot M^*$ given by $$\lambda(m^*)=\hbox{$\sum_i$} \lan m^*,n_{i[0]}\ran S^{-1}(n_{i[-1]})\ot n_i^*.$$ The definition of $\lambda$ can also be stated as follows: $\lambda(m^*)=m^*_{[-1]}\ot m^*_{[0]}$ if and only if $${\label{eq:5.2.1}}
\lan m_{[0]}^*,m\ran S(m_{[-1]}^*)=\lan m^*,m_{[0]}\ran m_{[-1]},$$ for all $m\in M$.

[\[pr:5.2\]]{} Let $M$ be a finitely generated projective left-left Yetter-Drinfeld module over the weak Hopf algebra $H$. Then $M^*$ with $H$-action and $H$-coaction given by [(\[eq:5.1.1\])]{} and [(\[eq:5.2.1\])]{} is also a left-left Yetter-Drinfeld module.

We have to show that $$\lambda(h\cdot m^*)=\hbox{$\sum_i$} \lan m^*,S(h)n_{i[0]}\ran S^{-1}(n_{i[-1]})\ot n_i^*$$ equals $$h_{(1)}m_{[-1]}^*S(h_{(3)})\ot h_{[2]}m_{[-1]}^*\\
= \hbox{$\sum_i$} \lan m^*,n_{i[0]}\ran h_{(1)}S^{-1}(n_{i[-1]})S(h_{(3)})\ot
(h_{(2)}\cdot n_i^*).$$ It suffices to show that both terms coincide after we evaluate the second tensor factor at an arbitrary $m\in M$. $$\begin{aligned}
&&\hspace*{-2cm}
\hbox{$\sum_i$} \lan m^*,n_{i[0]}\ran h_{(1)}S^{-1}(n_{i[-1]})S(h_{(3)})
\lan n_i^*,S(h_{(2)})m\ran\\
&=&\lan m^*,(S(h_{(2)})m)_{[0]}\ran h_{(1)}
S^{-1}\Bigl((S(h_{(2)})m)_{[-1]}\Bigr)S(h_{(3)})\\
&\equal{{(\ref{eq:2.2.1})}}&
\lan m^*, S(h_{(3)})m_{[0]}\ran h_{(1)}
S^{-1}\Bigl(S(h_{(4)})m_{[-1]}S^2(h_{(2)})\Bigr) S(h_{(5)})\\
&=& \lan m^*, S(h_{(3)})m_{[0]}\ran h_{(1)}
S(h_{(2)})S^{-1}(m_{[-1]})h_{(4)} S(h_{(5)})\\
&=& \lan m^*, S(h_{(2)})m_{[0]}\ran \varepsilon_t(h_{(1)})
S^{-1}(m_{[-1]})\varepsilon_t(h_{(3)})\end{aligned}$$ $$\begin{aligned}
&\equal{{(\ref{eq:1.6.3})}}&
\lan m^*, S(1_{(2)}h_{(1)})m_{[0]}\ran S(1_{(1)})
S^{-1}(m_{[-1]})\varepsilon_t(h_{(2)})\\
&=& \lan m^*, S(h_{(1)})1_{(1)}m_{[0]}\ran 1_{(2)}
S^{-1}(m_{[-1]})\varepsilon_t(h_{(2)})\\
&\equal{(\ref{eq:1.1.3},\ref{eq:2.3.1})}&
\lan m^*, S(1_{(1)}h)m_{[0]}\ran S^{-1}(m_{[-1]})1_{(2)}\\
&=& \lan m^*, S(h)S(1_{(1)})m_{[0]}\ran S^{-1}(S(1_{(2)})m_{[-1]})\\
&=& \lan m^*, S(h)1_{(2)}m_{[0]}\ran S^{-1}(1_{(1)}m_{[-1]})\\
&\equal{{(\ref{eq:2.1.1})}}&
\lan m^*, S(h)m_{[0]}\ran S^{-1}(m_{[-1]})\\
&=& \hbox{$\sum_i$}\lan m^*,S(h)n_{i[0]}\ran S^{-1}(n_{i[-1]}^*)\lan n_i^*,m\ran\end{aligned}$$

[\[pr:5.3\]]{} The category of finitely generated projective left-left Yetter-Drinfeld modules has left duality.

In view of the previous results, it suffices to show that the evaluation map ${\rm ev}_M$ and the coevaluation map ${\rm coev}_M$ are left $H$-colinear, for every finitely generated projective left-left Yetter-Drinfeld module $M$. Let us first show that ${\rm ev}_M$ is left $H$-colinear. $$\begin{aligned}
&&\hspace*{-2cm}
(H\ot {\rm ev}_M)(\lambda(1_{(1)}\cdot m^*\ot 1_{(2)}m))\\
&=& m^*_{[-1]}m_{[-1]}\ot \lan m^*_{[0]},1_{(1)}m_{[0]}\ran 1_{(2)}\\
&\equal{{(\ref{eq:5.2.1})}}&
\lan m^*,(1_{(1)}m_{[0]})_{[0]}\ran S^{-1}((1_{(1)}m_{[0]})_{[-1]})m_{[-1]}\ot 1_{(2)}\\
&\equal{{(\ref{eq:2.2bis.1})}}&
\lan m^*,m_{[0]}\ran 1_{(1)}S^{-1}(m_{[-1]})m_{[-2]}\ot 1_{(2)}\\
&=&\lan m^*,m_{[0]}\ran 1_{(1)}\varepsilon_t(S^{-1}(m_{[-1]}))\ot 1_{(2)}\\
&\equal{{(\ref{eq:2.3.1})}}&
\lan m^*,1_{(1')}m_{[0]}\ran 1_{(1)}\varepsilon_t(1_{(2')}S^{-1}(m_{[-1]}))\ot 1_{(2)}\\
&\equal{{(\ref{eq:1.1.7})}}&
\lan m^*,1_{(1')}m_{[0]}\ran 1_{(1)}1_{(2')}\varepsilon_t(S^{-1}(m_{[-1]}))\ot 1_{(2)}\\
&\equal{{(\ref{eq:1.9.2})}}&
\lan m^*,1_{(1')}S^{-1}(\varepsilon_t(S^{-1}(m_{[-1]})))m_{[0]}\ran 1_{(1)}1_{(2')}\ot 1_{(2)}\\
&\equal{{(\ref{eq:1.7.1})}}&
\lan m^*,1_{(1')}\varepsilon_s(S^{-2}(m_{[-1]}))m_{[0]}\ran 1_{(1)}1_{(2')}\ot 1_{(2)}\\
&\equal{(\ref{eq:1.1.1},\ref{eq:2.3.2})}&
\lan m^*,1_{(1)}m\ran 1_{(2)}\ot 1_{(3)}\equal{{(\ref{eq:2.5.2})}}
\lambda(\lan m^*,1_{(1)}m\ran 1_{(2)})\\
&=&\lambda({\rm ev}_M(1_{(1)}\cdot m^*\ot 1_{(2)}m)).\end{aligned}$$ To prove that ${\rm coev}_M$ is left $H$-colinear, we have to show that, for all $z\in H_t$, $$\begin{aligned}
&&\hspace*{-2cm}
\lambda({\rm coev}_M(z))=\hbox{$\sum_i$} \lambda(1_{(1)}zn_i\ot 1_{(2)}\cdot n_i^*)\\
&=&
\hbox{$\sum_i$} (1_{(1)}zn_i)_{[-1]}(1_{(2)}\cdot n_i^*)_{[-1]}
\ot (1_{(1)}zn_i)_{[0]}\ot (1_{(2)}\cdot n_i^*)_{[0]}\end{aligned}$$ equals $$(H\ot {\rm coev}_M)(\lambda(z))=
(H\ot {\rm coev}_M)(1_{(1)}z\ot 1_{(2)})=
 \hbox{$\sum_i$} 1_{(1)}z\ot 1_{(2)}n_i\ot 1_{(3)}\cdot n_i^*.$$ It suffices to show that both terms coincide after we evaluate the third tensor factor at an arbitrary $m\in M$. Indeed $$\begin{aligned}
&&\hspace*{-2cm}
\hbox{$\sum_i$} (1_{(1)}zn_i)_{[-1]}(1_{(2)}\cdot n_i^*)_{[-1]}
\ot (1_{(1)}zn_i)_{[0]}\lan (1_{(2)}\cdot n_i^*)_{[0]},m\ran\\
&\equal{{(\ref{eq:5.2.1})}}&
\hbox{$\sum_i$} (1_{(1)}zn_i)_{[-1]} \lan 1_{(2)}\cdot n_i^*, m_{[0]}\ran S^{-1}(m_{[-1]})
\ot (1_{(1)}zn_i)_{[0]}\\
&=&
\hbox{$\sum_i$} (1_{(1)}zn_i)_{[-1]} \lan n_i^*, S(1_{(2)})m_{[0]}\ran S^{-1}(m_{[-1]})
\ot (1_{(1)}zn_i)_{[0]}\\
&=& (1_{(1)}zS(1_{(2)})m_{[0]})_{[-1]} S^{-1}(m_{[-1]}) \ot (1_{(1)}zS(1_{(2)})m_{[0]})_{[0]} \end{aligned}$$ $$\begin{aligned}
&\equal{(\ref{eq:1.1.4},\ref{eq:2.2.1})}&
1_{(1)}zm_{[-1]}S(1_{(3)})S^{-1}(m_{[-2]}) \ot 1_{(2)}m_{[0]}\\
&=&
1_{(1)}zm_{[-1]}S(1_{(2')})S^{-1}(m_{[-2]}) \ot 1_{(2)}1_{(1')}m_{[0]}\\
&\equal{{(\ref{eq:2.2.2})}}&
1_{(1)}zm_{[-1]}S^{-1}(m_{[-2]}) \ot 1_{(2)}m_{[0]}\\
&=&
1_{(1)}zS^{-1}(\varepsilon_t(m_{[-1]})) \ot 1_{(2)}m_{[0]}\\
&\equal{{(\ref{eq:1.9.2})}}&
1_{(1)}z \ot 1_{(2)}\varepsilon_t(m_{[-1]})m_{[0]}
\equal{{(\ref{eq:2.1.3})}}1_{(1)}z \ot 1_{(2)}m\\
&=& 1_{(1)}z \ot 1_{(2)}S(1_{(3)})m=
\hbox{$\sum_i$} 1_{(1)}z \ot 1_{(2)}n_i\lan n_i^*,S(1_{(3)})m\ran\\
&=& \hbox{$\sum_i$} 1_{(1)}z \ot 1_{(2)}n_i\lan 1_{(3)}\cdot n_i^*,m\ran,\end{aligned}$$ as needed.

Appendix. Weak bialgebras and bialgebroids
==========================================

[\[se:6\]]{} In [@Schauenburg0], Yetter-Drinfeld modules over a $\times_R$-bialgebra (see [@Takeuchi]) are introduced, and it is shown that the weak center of the category of left modules is isomorphic to the category of Yetter-Drinfeld modules. The notion of $\times_R$-bialgebra is equivalent to the notion of $R$-bialgebroid, we refer to [@BM] for a detailed discussion. So we can consider Yetter-Drinfeld modules over bialgebroids.\
A weak bialgebra $H$ can be viewed as a bialgebroid over the target space $H_t$; this was shown in [@EN] in the weak Hopf algebra case, and generalized to the weak bialgebra case in [@Schauenburg]. The aim of this Section is to make clear that Yetter-Drinfeld modules over $H$ considered as a weak bialgebra coincide with Yetter-Drinfeld modules over $H$-considered as a bialgebroid.\
To this end, we first recall the definition of a bialgebroid, as introduced by Lu [@Lu]. Let $k$ be a commutative ring, and $R$ a $k$-algebra. An $R\ot R^{\rm op}$-ring is a pair $(H,i)$, with $H$ a $k$-algebra and $i:\ R\ot R^{\rm op}\to H$. Giving $i$ is equivalent to giving algebra maps $s_H:\ R\to H$ and $t_H:\ R\to H^{\rm op}$ satisfying $s_H(a)t_H(b)=t_H(b)s_H(a)$, for all $a,b\in R$. We then have that $i(a\ot b)=s_H(a)t_H(b)$. Restriction of scalars makes $H$ into a left $R\ot R^{\rm op}$-module, and an $R$-bimodule: $$a\cdot h\cdot b=s_H(a)t_H(b)h.$$ Consider $$\begin{aligned}
&&\hspace*{-2cm}
H\times_R H=\{\sum_i h_i\ot_R k_i\in H\ot_R H\\
&|&
\sum_i h_it_H(a)\ot_R k_i=\sum_i h_i\ot_R k_is_H(a),~~{\rm for~all~}a\in R\}\end{aligned}$$ It is easy to show that $H\times_R H$ is a $k$-subalgebra of $H\ot_R H$.\
Recall that an $R$-coring is a triple $(H,\tilde{\Delta},\tilde{\varepsilon})$, with $H$ an $R$-bimodule and $\tilde{\Delta}:\ H\to H\ot_R H$ and $\tilde{\varepsilon}:\ H\to R$ $R$-bimodule maps satisfying the usual coassociativity and counit properties; we refer to [@BW] for a detailed discussion of corings.

[\[de:6.1\]]{} [@Lu] A left $R$-bialgebroid is a fivetuple $(H,s_H,t_H,\tilde{\Delta},\tilde{\varepsilon})$ satisfying the following conditions.

1.  $(H,\tilde{\Delta},\tilde{\varepsilon})$ is an $R$-coring;

2.  $(H,m\circ (s_H\ot t_H)=i)$ is an $R\ot R^{\rm op}$-ring;

3.  ${{\rm Im}\,}(\tilde{\Delta})\subset H\times_R H$;

4.  $\tilde{\Delta}:\ H\to H\times_R H$ is an algebra map, $\tilde{\varepsilon}(1_H)=
    1_R$ and $$\tilde{\varepsilon}(gh)=\tilde{\varepsilon}(gs_H(\tilde{\varepsilon}(h)))=
    \tilde{\varepsilon}(gt_H(\tilde{\varepsilon}(h))),$$ for all $g,h\in H$.

Take two left $H$-modules $M$ and $N$; then $M$ and $N$ are $R$-bimodules, by restriction of scalars. $M\ot_RN$ is a left $H$-module, with $$h\cdot (m\ot_Rn)=h_{(1)}m\ot_R h_{(2)}n.$$ Also $R$ is a left $H$-module, with $$h\cdot r=\tilde{\varepsilon}(hs_H(r))=\tilde{\varepsilon}(ht_H(r)).$$ $({}_H{\mathcal{M}},\ot_R,R)$ is a monoidal category, and the restriction of scalars functor ${}_H{\mathcal{M}}\to {}_R{\mathcal{M}}_R$ is strictly monoidal; this can be used to reformulate the definition of a bialgebroid (see [@BM; @Schauenburg98; @Szlachanyi]).\
In [@Schauenburg0 Sec. 4], left-left Yetter-Drinfeld modules over $H$ are introduced, and it is shown that ${\mathcal{W}}_l({}_H{\mathcal{M}})$ is isomorphic to the category of Yetter-Drinfeld modules. According to [@Schauenburg0], a left-left Yetter-Drinfeld $H$-module is a left comodule $M$ over the coring $H$, together with a left $H$-action on $M$ such that the underlying left $R$-actions coincide, and such that $${\label{eq:6.1.1}}
h_{(1)}m_{[-1]}\ot_R h_{(2)}\cdot m_{[0]}=(h_{(1)}\cdot m)_{[-1]}h_{(2)}
\ot_R (h_{(1)}\cdot m)_{[0]}$$ holds in $H\ot_R M$, for all $h\in H$ and $m\in M$.\
Let $H$ be a weak bialgebra, and consider the maps $$\begin{aligned}
&&\hspace*{-2cm} s_H:\ H_t\rTo^{\subset} H;\\
&&\hspace*{-2cm} t_H=\ol{\varepsilon}_{s|H_t}:\ H_t\to H_s\subset H;\\
&&\hspace*{-2cm} \tilde{\Delta}={\rm can}\circ \Delta:\
H\to H\ot H\rTo^{\rm can}H\ot_{H_t}H;\\
&&\hspace*{-2cm} \tilde{\varepsilon}=\varepsilon_t:\ H\to H_t.\end{aligned}$$ Then $(H,s_H,t_H,\tilde{\Delta},\tilde{\varepsilon})$ is a left $H_t$-bialgebroid. The fact that ${{\rm Im}\,}(\tilde{\Delta})\subset H\times_{H_t}H$ follows from the separability of $H_t$ as a $k$-algebra (cf. [Proposition \[pr:1.2b\]]{}).\
We have seen in [Section \[se:1.1\]]{} that, for any two left $H$-modules $M$ and $N$, we have an isomorphism $\ol{\pi}:\ M\ot_{H_t}N\to M\ot_t N$. This entails that the monoidal categories $({}_H{\mathcal{M}},\ot_t,H_t)$ and $({}_H{\mathcal{M}},\ot_{H_t},H_t)$ are isomorphic, and a fortiori, their weak left centers are isomorphic categories. Consequently, the two corresponding categories of Yetter-Drinfeld modules are isomorphic. This can also be seen directly, comparing the definitions in [Section \[se:2\]]{} and [(\[eq:6.1.1\])]{}.

[**Acknowledgment**]{}

We thank Tomasz Brzeziński and the referee for their useful comments, and Adriana Nenciu for sending us her paper [@Nenciu].

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[^1]: Research supported by the projects G.0278.01 “Construction and applications of non-commutative geometry: from algebra to physics" from FWO-Vlaanderen and “New computational, geometric and algebraic methods applied to quantum groups and differential operators" from the Flemish and Chinese governments.
