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\begin{document}

\title[DRINFEL'D DOUBLES AND EHRESMANN DOUBLES]{Drinfel'd doubles
and Ehresmann doubles for\\ Lie algebroids and Lie bialgebroids}

\author{K. C. H. Mackenzie}
\address{School of Mathematics and Statistics, %\\
	   University of Sheffield, %\\
	   Sheffield, S3 7RH, %\\
	   England}

\email{K.Mackenzie@sheffield.ac.uk}

\issueinfo{4}{11}{}{1998}
\dateposted{October 22, 1998}
\pagespan{74}{87}
\PII{S 1079-6762(98)00050-X}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}

\subjclass{Primary 58F05; Secondary 17B66, 18D05, 22A22, 58H05} %.}
\date{July 12, 1998}

\commby{Frances Kirwan}

%\keywords{}
\begin{abstract}
We show that the Manin triple characterization of Lie bialgebras in terms
of the Drinfel'd double may be extended to arbitrary Poisson manifolds and
indeed Lie bialgebroids by using double cotangent bundles, rather than the
direct sum structures (Courant algebroids) utilized for similar purposes by
Liu, Weinstein and Xu. This is achieved in terms of an abstract notion of
double Lie algebroid (where \lq\lq double\rq\rq\ is now used in the
Ehresmann sense) which unifies many iterated constructions in differential
geometry.
\end{abstract}

\maketitle

\section{Introduction}

It is well known that Lie bialgebras may be characterized in terms of
Manin triples: a Lie algebra ${\mathfrak g}$ with a Lie algebra structure on
the dual ${\mathfrak g}^*$ is a Lie bialgebra if and only if the vector
space direct sum ${\mathfrak g}\oplus{\mathfrak g}^*$ has a Lie algebra structure
making ${\mathfrak g}$ and ${\mathfrak g}^*$ isotropic subalgebras and making the
natural pairing of ${\mathfrak g}\oplus{\mathfrak g}^*$ with itself invariant
\cite{Drinfeld:1983}. For a general Poisson manifold $P$, both $TP$ and $T^*P$
have Lie algebroid structures and it is reasonable to seek a version of
the Manin triple result for this case and, more generally, for Lie
bialgebroids in the sense of the author and Xu \cite{MackenzieX:1994}.

One can quickly see that the direct transposition of the notion of Manin
triple to the Lie bialgebroid $(TP, T^*P)$ is only valid under very
restrictive conditions on $P$. Very recently Liu, Weinstein and Xu
\cite{LiuWX:1997} have given a highly nontrivial development of the notion
of Manin triple to the bialgebroid case. For a Lie bialgebroid $(A, A^*)$
they define a bracket structure on $\Ga(A\oplus A^*)$ which is recognizably
an extension of the Manin formula and which, though it is not itself
usually a Lie algebroid bracket, induces the given brackets on $\Ga A$
and $\Ga A^*$. The properties of this bracket on $E = A\oplus A^*$ are
abstracted by \cite{LiuWX:1997} into a notion called {\em Courant
algebroid}, and those Courant algebroids $E$ which arise from Lie
bialgebroids are characterized as those which possess a transversal pair
of integrable isotropic subbundles $L_1, L_2\subseteq E$; these are
then $A, A^*$ for a Lie bialgebroid structure.

We argue here that it is also valid, and in certain senses more natural, to
regard the double of a Poisson manifold $P$ (by which we mean the double of
its Lie bialgebroid) as $T^*TP\cong T^*T^*P$ rather
than $TP\oplus T^*P$. More generally, we provide a detailed justification
for regarding the double of a Lie bialgebroid $(A, A^*)$ as not
$A\oplus A^*$ but $T^*A\cong T^*A^*$. Here $T^*A$ should be regarded
as in \cite[\S5]{MackenzieX:1994} as the double vector bundle in
Figure~\ref{fig:T*A},
\begin{figure}[htb]
\begin{picture}(360,90)
\put(120,40){$\begin{matrix}&&      &&\\
        &T^*A&\mlra&A^*&\\
        &&&&\\
        &\Bigg\downarrow&&\Bigg\downarrow&\\
        &&&&\\
        &A&\mlra&M&\\
        &&&&
\end{matrix}$}
\end{picture}
\caption{\ \label{fig:T*A}}
\end{figure}
where $T^*A\to A$ is the usual cotangent bundle and $T^*A\to A^*$ is the
composition of $T^*A^*\to A^*$ with the canonical antisymplectomorphism
$R^{-1}\colon T^*A\to T^*A^*$. If $A$ is written locally as $M\times V$,
then $R\colon T^*M\times V^*\times V\to T^*M\times V\times V^*$ interchanges
$V$ with $V^*$, and is $-\myid$ on $T^*M$; see \cite[5.5]{MackenzieX:1994}.
We call Figure~\ref{fig:T*A} the {\em double cotangent bundle of} $A$.
If $A$ is a Lie algebroid, then $A^*$ has its dual Poisson structure and
this induces a Lie algebroid structure on $T^*A^*\to A^*$. Likewise a
Lie algebroid structure on $A^*$ induces a Lie algebroid structure on
$T^*A\to A$. Thus Lie algebroid structures on $A$ and $A^*$ equip the
four sides of Figure~\ref{fig:T*A} with Lie algebroid structures. What
we are seeking is an abstract set of compatibility conditions for these
structures which, when applied to Figure~\ref{fig:T*A}, will characterize
Lie bialgebroid stuctures on $A$. This is our first problem.

The key to this is the notion of double Lie groupoid in the sense of a Lie
groupoid object in the category of Lie groupoids. This sense of the word
\lq\lq double\rq\rq\ goes back to Ehresmann \cite[III--1]{Ehresmann};
double and multiple structures in this sense have been used for many years
in homotopy theory \cite{Brown:1987}. This use of the word should be
carefully distinguished from the Drinfel'd sense \cite{Drinfeld:1987},
which we have been discussing until now, in which a double is a self-dual
object composed from a fundamental structure and its dual.
Nonetheless it is one consequence of this work that the two notions of
double correspond in a sense which we will make precise.

In fact the presence of double groupoids underlying Poisson groups is
already well established. Lu and Weinstein \cite{LuW:1989} proved that a
Poisson group may always be integrated to a symplectic double groupoid
and, slightly earlier, the same authors \cite{LuW:1990} and Majid
\cite{Majid:1990} gave sufficent conditions for the dressing transformations
of a Poisson Lie group to integrate globally; it was then shown in
\cite[\S\S2, 4]{Mackenzie:1992} that the twisted multiplicativity
conditions of these global actions reflect exactly the fact that they come
from a double groupoid structure. See \S\ref{sect:dlg} below.

The author's interest in a prospective Lie theory for double groupoids
goes back to 1988, before he was properly aware of the relationships
between Poisson geometry and groupoid and algebroid theory. As stated
in the Introduction to \cite{Mackenzie:1992}, the principal purpose was
to understand higher-order constructions in connection theory and more
generally in differential geometry, specifically for rendering natural
the apparent lack of structure on jet prolongation spaces by embedding
them in iterated multiple structures of the type encompassed by the
theory given here. This aspect is still of interest and will be treated
elsewhere.

In order to proceed, we need to describe the notion of double Lie groupoid
in more detail (see \cite{Mackenzie:1992} and references given there).
A double Lie groupoid consists of a manifold $S$ equipped with two Lie
groupoid structures on bases $H$ and $V$, each of which is a Lie
groupoid over a common base $M$, such that the structure maps (source,
target, multiplication, identity, inversion) of each groupoid structure
on $S$ are morphisms with respect to the other; see Figure~\ref{fig:S}.
One should think of elements of $S$ as squares, the horizontal edges of
which come from $H$, the vertical edges from $V$, and the corner points
from $M$. Since $S$ is a Lie groupoid over $H$ we may take its Lie algebroid,
called $A_VS$, and because the Lie functor preserves diagrams and pullbacks,
$A_VS$ is a Lie groupoid over $AV$; see Figure~\ref{fig:vLAgpd}(a).
These two structures on $A_VS$ commute in the same way that the original
groupoid structures on $S$ commute: each of the groupoid structure maps
is a morphism of Lie algebroids. We call structures of this type
\LAgpd s. In the case of a double groupoid arising from a Poisson group
$G$ with dual group $G^*$, the \LAgpd s are $T^*G$ and $T^*G^*$.
See \cite[\S4]{Mackenzie:1992}.

Now one may take the Lie algebroid of the horizontal groupoid structure on
$A_VS$ and obtain a double vector bundle $A^2S = A(A_VS)$  whose horizontal
structure is a Lie algebroid. Interchanging the order of the processes---so
that one first takes the Lie algebroid of the horizontal structure on $S$
and then the Lie algebroid of the resulting vertical structure---one obtains
a second double vector bundle $A_2S = A(A_HS)$ with the Lie algebroid
structure now placed vertically. These two double vector bundles may be
identified by a map derived from the canonical involution in the double
tangent bundle of $S$, and one thus obtains a double vector bundle all four
sides of which have Lie algebroid structures; this is what we call the
double Lie algebroid of $S$. It includes, amongst many examples, the double
tangent bundle of an arbitrary manifold, the Lie bialgebra of a Poisson
group, and the double cotangent bundle of a Poisson manifold. See
\cite{Mackenzie:Doubla2}.

The construction of $A^2S \cong A_2S$ from $S$ ensures that the four
Lie algebroid structures on the sides of Figure~\ref{fig:vLAgpd}(b)
are suitably compatible. Our second problem is to isolate these
compatibility conditions and give a general definition of a double
Lie algebroid without reference to any underlying double groupoid.

We will see that the solutions to these two problems are closely
related.

The difficulty is that whereas the notions of double groupoid and of
\LAgpd\ can be defined by morphism conditions on the structure maps
of a groupoid, this is not possible with double Lie algebroids. The
bracket of a Lie algebroid is defined not on the bundle itself but on
its module of sections and it is not clear what could be meant by
requiring one bracket to be a morphism with respect to the other.
Instead we proceed by an indirect route, which makes essential use of
the duality between Lie algebroids and (fibrewise linear) Poisson
structures. Thus summarized, this strategy may sound straightforward,
but its implementation is not: the duals to be taken are duals of double
and triple structures and have many unexpected features.

This is a summary and overview of \cite{Mackenzie:Doubla2},
\cite{Mackenzie:SDGDPG}, and \cite{Mackenzie:DLADLB}, together with
\cite{Mackenzie:1992}. I am grateful to many people for comments on
various stages of the work, acknowledged in the individual papers,
and to Yvette Kosmann-Schwarzbach also for comments on this announcement.
Overall I especially want to thank Ronnie Brown, Alan Weinstein, and Ping Xu.

\section{Double Lie groupoids and associated Lie algebroids}
\label{sect:dlg}

It is valuable to keep in mind the very classical examples of the
double (or iterated) tangent bundle $T^2M$ of a manifold $M$, as in
Figure~\ref{fig:T2M}(a), and more generally the tangent bundle $TA$ of
an arbitrary vector bundle $(A,q,M)$, as in Figure~\ref{fig:T2M}(b).

\begin{figure}[htb]
\begin{picture}(360,100)
\put(0,40){$\begin{matrix}&&T(p)  &&\cr
        &T^2M&\lrah&TM&\cr
        &&&&\cr
 p_{TM} &\Bigg\downarrow&&\Bigg\downarrow&p\cr
        &&&&\cr
        &TM&\lrah&M&\cr
        &&p&&\cr
        &&&&\cr
        &&\mbox{(a)}&&\cr
\end{matrix}$}
%
\put(190,40){$\begin{matrix}&&T(q)  &&\cr
        &TA&\lrah&TM&\cr
        &&&&\cr
     p_A&\Bigg\downarrow&&\Bigg\downarrow&p\cr
        &&&&\cr
        &A&\lrah&M&\cr
        &&q&&\cr
        &&&&\cr
        &&\mbox{(b)}&&\cr
\end{matrix}$}
\end{picture}
\caption{\ \label{fig:T2M}}
\end{figure}
As well as its standard tangent bundle structure over $A$, there is a
{\em prolonged} vector bundle structure $(TA, T(q), TM)$ obtained by
applying the tangent functor to the bundle projection, the zero
section, the addition, and the scalar multiplication, of $A$. This is a
{\em double vector bundle} in the sense of \cite{Pradines:DVB},
\cite[\S1]{Mackenzie:1992}; that is, the structure maps of each structure
on $TA$ are morphisms with respect to the other structure. For a traditional
treatment, see \cite{Besse}.

Recalling the definition sketched in the Introduction, we picture a
double Lie groupoid $(S;H,V;M)$ and a typical element as in
Figure~\ref{fig:S};
\begin{figure}[htb]
\begin{picture}(360,100)
\put(10,40){$\begin{matrix}&&{\tilalpha_H,\tilbeta_H}&&\cr
          &S&\lgpd &V&\cr
          &&&&\cr
          {\tilalpha_V,\tilbeta_V}&\vgpd&&\vgpd&{\alpha_V,\beta_V}\cr
          &&&&\cr
          &H&\lgpd &M&\cr
          &&{\alpha_H,\beta_H}&&
\end{matrix}$}
\put(220,40){$\begin{matrix}&\raise.7ex\hbox{$\tilbeta_V(s)$}&\cr
                  \raise.35in\hbox{$\tilbeta_H(s)$}&\ssq&\raise.35in
                         \hbox{$\tilalpha_H(s)$}\cr
                         &\tilalpha_V(s)&
\end{matrix}$}
\put(280,40){$s$}
\end{picture}
\caption{\ \label{fig:S}}
\end{figure}
%
here $\alpha$ and $\beta$ generically denote source and target projections
respectively.

The definition ensures that when a composite of four elements as in
Figure~\ref{fig:squares}
\begin{figure}[htb]
\begin{picture}(360,120)(35,0)
\put(160,65){$\ssq$}                    \put(230,65){$\ssq$}
\put(180,85){$s_2$}                     \put(255,85){$s_1$}
\put(160,0){$\ssq$}                     \put(230,0){$\ssq$}
\put(180,25){$s_3$}                     \put(255,25){$s_4$}
\end{picture}\caption{\ \label{fig:squares}}
\end{figure}
may be calculated as $(s_2\hcomp s_1)\vcomp(s_3\hcomp s_4)$, that is, first
composing horizontally and then vertically, then it may also be
calculated as $(s_2\vcomp s_3)\hcomp(s_1\vcomp s_4)$, and the results are
equal. The smoothness condition ensures (see \cite[1.2]{BrownM:1992})
that both structures on $S$ are Lie groupoids (called differentiable
groupoids in \cite{Mackenzie:LGLADG}) and that the Lie functor may be
applied to the structure maps. We will describe examples below.

Applying the Lie functor to the vertical structure on $S$ we obtain
Figure~\ref{fig:vLAgpd}(a).
\begin{figure}[hbt]
\begin{picture}(360,100)
\put(0,40){$\begin{matrix} && A(\tilalpha_H), A(\tilbeta_H)  &&\cr
                            &A_VS    &\lgpd &AV&\cr
                            &&&&\cr
                    \tilq_V &\Bigg\downarrow&&\Bigg\downarrow&q_V\cr
                            &&&&\cr
                            &H &\lgpd &M&\cr
                            &&&&\cr
                            &&&&\cr
                            &&\mbox{(a)}
		 \end{matrix}$}
\put(190,40){$\begin{matrix}&&\ts{q}_H&&\cr
        &A^2S&\lrah   &AV&\cr
        &&&&\cr
A(\tilq_V) &\Bigg\downarrow& &\Bigg\downarrow&q_V\cr
        &&&&\cr
        &AH&\lrah &M&\cr
        &&q_H&&\cr
        &&&&\cr
        &&\mbox{(b)}&&
	  \end{matrix}$}
\end{picture}
\caption{\ \label{fig:vLAgpd}}
\end{figure}
Since the Lie functor preserves pullbacks, $A_VS$, the Lie algebroid of
$S\gpd H$, has a prolonged Lie groupoid structure over $AV$. See
\cite[\S4]{Mackenzie:1992}. Now $A^2S = A(A_VS)$ in (b) is the Lie algebroid
of this Lie groupoid. Using the Lie functor in the same way as the tangent
functor is used in Figure~\ref{fig:T2M}, $A^2S$ has a vector bundle structure
over $AH$. Thus $A^2S$ is a double vector bundle with Lie algebroid
structures on three of its four sides. See \cite[\S2]{Mackenzie:Doubla2}.

We can likewise take first the horizontal Lie algebroid, obtaining a
Lie groupoid $A_HS\gpd AH$ as in Figure~\ref{fig:hLAgpd}(a), and then
define $A_2S = A(A_HS)$ as in Figure~\ref{fig:hLAgpd}(b).

\begin{figure}[hbt]
\begin{picture}(360,100)
\put(20,40){$\begin{matrix} &&\tilq_H &&\cr
                            &A_HS    &\mlra &V &\cr
                            &&&&\cr
                            &\vgpd&&\vgpd&\cr
                            &&&&\cr
                            &AH &\mlra &M&\cr
                            &&&&\cr
                            &&&&\cr
                            &&\mbox{(a)}&&
                            \end{matrix}$}
\put(0,55){$\scriptstyle{A(\tilbeta_V)}$}
\put(0,45){$\scriptstyle{A(\tilalpha_V)}$}
%
\put(190,40){$\begin{matrix}&&A(\tilq_H)&&\cr
        &A_2S&\lrah &AV&\cr
        &&&&\cr
\ts{q}_V&\Bigg\downarrow& &\Bigg\downarrow&q_V\cr
        &&&&\cr
        &AH&\lrah &M&\cr
        &&q_H&&\cr
        &&&&\cr
        &&\mbox{(b)}&&
        \end{matrix}$}
\end{picture}
\caption{\ \label{fig:hLAgpd}}
\end{figure}

Now the canonical involution $J_S\colon T^2S\to T^2S$ restricts to give
an isomorphism of double vector bundles $j_S\colon A^2S\to A_2S$. Using
this we transport the Lie algebroid structure of $A_2S\to AH$ back to
give $A^2S$ four Lie algebroid structures. Thus equipped, $A^2S$ is the
{\em double Lie algebroid} of $(S;H,V;M)$. See \cite[\S2]{Mackenzie:Doubla2}.

\begin{example}
Taking $H = V = M\times M$ to be the pair groupoid, and $S = M^4$ to be
the double groupoid whose elements are the four corners of empty squares,
we have that $A_VS$ is the pair groupoid $TM\times TM$, and so $A^2S = T^2M$,
with horizontal structure the tangent of $TM$ and vertical structure the
tangent prolongation of $TM\to M$. Likewise $A_2S$ is $T^2M$ with the
horizontal and vertical structures interchanged.

More generally, take $S = G\times G$ where $G\gpd M$ is any Lie groupoid
and $S$ is horizontally the pair groupoid over $G$ and vertically the
Cartesian square groupoid over $M\times M$. Then $A_HS = TG$ is the tangent
groupoid over $TM$ and $A_2S = ATG$, the Lie algebroid of the tangent
groupoid. On the other hand, $A_VS = AG\times AG$, the pair groupoid,
and $A^2S = TAG$ the tangent prolongation, defined intrinsically in
\cite[\S5]{MackenzieX:1994}, of the Lie algebroid $AG\to M$. The
canonical isomorphism $j_G\colon TAG\to ATG$ is described in
\cite[\S7]{MackenzieX:1994}.
\end{example}

An important class of examples presents naturally as structures intermediate
between double Lie groupoids and double Lie algebroids. An {\em \LAgpd}\
\cite[\S4]{Mackenzie:1992} $(\Om;G,A;M)$, as in Figure~\ref{fig:Om}(a),
comprises both a Lie groupoid structure on $\Om$ over base $A$ (which is
itself a Lie algebroid over $M$), and a Lie algebroid on $\Om$ over $G$
(which is a Lie groupoid over $M$); the two structures on $\Om$ again are
to commute in the sense that the maps defining the groupoid structure are
all Lie algebroid morphisms (see \cite{HigginsM:1990}).
\begin{figure}[hbt]
\begin{picture}(360,100)
\put(20,40){$\begin{matrix} &&   \tilalpha, \tilbeta  &&\cr
                            &\Omega &\sgpd &A&\cr
                            &&&&\cr
                    \tilq   &\Bigg\downarrow&&\Bigg\downarrow&q  \cr
                            &&&&\cr
                            &G &\sgpd &M&\cr
                            &&&&\cr
                            &&&&\cr
                            &&\mbox{(a)}&&
                            \end{matrix}$}
%
\put(200,40){$\begin{matrix}&& \ts{q} &&\cr
        &A\Omega&\lrah   &A&\cr
        &&&&\cr
A(\tilq) &\Bigg\downarrow& &\Bigg\downarrow&q\cr
        &&&&\cr
        &AG&\lrah &M&\cr
        &&q_G&&\cr
        &&&&\cr
        &&\mbox{(b)}&&
        \end{matrix}$}
\end{picture}
\caption{\ \label{fig:Om}}
\end{figure}
The structures $A_VS$ and $A_HS$ associated with any double Lie groupoid are
\LAgpd s. If the Lie algebroid structures have zero bracket and zero anchor---in other
words are vector bundles---then we speak of a {\em \VBgpd}.

Consider a Lie group $G$ equipped with an arbitrary Poisson structure
$\pi^\#\colon T^*G\to TG$. Then $T^*G$ has a Lie groupoid structure on base
${\mathfrak g}^*$ arising from the Lie group structure \cite{CosteDW} and
$T^*G\to G$ has a Lie algebroid structure arising from the Poisson
structure. Now $(G,\pi)$ is a Poisson Lie group if and only if these
two structures make $T^*G$ an \LAgpd; indeed it is a Poisson Lie group if
and only if $\pi^\#$ is a groupoid morphism \cite[4.12]{Mackenzie:1992}.

More generally, any Lie groupoid $G\gpd M$ gives rise to its cotangent
groupoid $T^*G\gpd A^*G$ (\cite{CosteDW}, \cite{Pradines:1988}) and this
is a \VBgpd\ $(T^*G;G,A^*G;M)$. If $G$ is equipped with a Poisson
structure $\pi$, then $(G,\pi)$ is a Poisson groupoid as defined by Weinstein
\cite{Weinstein:1988} if and only if $\pi^\#\colon T^*G\to TG$ is a Lie
groupoid morphism (\cite{AlbertD:1991ssr}, \cite[8.1]{MackenzieX:1994}), and
again this is equivalent to the condition that $T^*G$ be an \LAgpd\ with
respect to the two structures.

This characterization of Poisson Lie group(oid)s is an irreducible and
nonobvious fact which shows that double structures in the Ehresmann
sense---the sense of this section---are fundamentally associated with
Poisson group theory. We will see further confirmation of this point
below.

In \cite[\S5]{MackenzieX:1994} it was shown that the structure of an
arbitrary Lie algebroid $A\to M$ can be prolonged to the tangent
prolongation $TA\to TM$; identifying the dual of $TA\to TM$ with
$T(A^*)\to TM$ \cite[5.3]{MackenzieX:1994}, the dual of this structure
is the tangent lift of the Poisson structure on $A^*$ dual to the Lie
algebroid structure of $A$.

More generally, consider an arbitrary \LAgpd\ as in Figure~\ref{fig:Om}(a).
Applying the Lie functor as in Figure~\ref{fig:vLAgpd}, we get the double
vector bundle in Figure~\ref{fig:Om}(b), and the vertical Lie algebroid
structure in Figure~\ref{fig:Om}(a) can be prolonged to the vertical
bundle of (b) \cite[\S1]{Mackenzie:Doubla2}.

In the case of $\Om = T^*G$ for $G$ a Poisson groupoid, we obtain
$A\Om = AT^*G$. This is known, by a general result for symplectic
groupoids (\cite{CosteDW}, or \cite[7.3]{MackenzieX:1994}), to be
isomorphic to $T^*A^*G$, the cotangent Lie algebroid of the Lie
algebroid dual.

In the case of the \LAgpd s of a double Lie groupoid $S$, we have the
following result.

\begin{theorem}                                           
%{\em 
[{\cite[2.3]{Mackenzie:Doubla2}}] \label{thm:j}
Let $(S;H,V;M)$ be a double Lie groupoid. Then $j_S$, regarded as a morphism
of vector bundles over $AH$, is an isomorphism of Lie algebroids from the
prolonged structure on $A(A_VS)\to AH$ to the Lie algebroid of $A_HS\gpd AH$.
\end{theorem}

The proof requires a substantial calculus which extends the classical
complete and vertical lifting processes for vector fields. See
\cite{MackenzieX:1998} for the case $S = G\times G$, which is
intermediate between the classical theory and the full proof.

Returning to the case of Poisson Lie groups, let $G^*$ be a Lie group
integrating ${\mathfrak g}^*$ and suppose that the coadjoint action of $G$ on
${\mathfrak g}^*$ and the dressing transformation action of ${\mathfrak g}^*$ on
$G$ integrate globally (for simple topological conditions guaranteeing
this, see \cite{LuW:1989}, \cite{Majid:1990}) to a left action
$G\times G^*\to G^*,\ (g,\phi)\mapsto {}^g\phi,$ and a right action
$G\times G^*\to G,\ (g,\phi)\mapsto g^\phi.$ Then $G\times G^*$ has a
double Lie groupoid structure with sides $G$ and $G^*$ defined by
regarding $(g, \phi)$ as the empty square
\begin{equation}                               \label{diag:vac}
\begin{matrix}&\raise.7ex\hbox{$g$}&\cr
                  \raise.35in\hbox{${}^g\phi$}&\ssq&\raise.35in
                         \hbox{$\phi$}\cr
                         &g^\phi&
\end{matrix}
\end{equation}
with compositions forced by Figure~\ref{fig:S}. Both groupoid structures
on $S = G\times G^*$ are action (or transformation) groupoids and
$A_VS = G\times{\mathfrak g}^* \isom T^*G$ as \LAgpd s. Dually,
$A_HS \isom T^*G^*$ as \LAgpd s. See \cite[\S\S2, 4]{Mackenzie:1992}.

Any double groupoid $(S;H,V;M)$ in which a square is determined by
two touching sides---a condition called {\em vacancy} in
\cite{Mackenzie:1992}---must be of this form: there are actions of $H$ on
$V$ and of $V$ on $H$ which obey twisted multiplicativity conditions like
those of integrated dressing transformations, and the two structures on
$S$ are the corresponding action groupoids. Thus the twisted equations
characteristic of dressing transformations emerge precisely from the double
groupoid conditions and the simple further condition of vacancy. See
\cite[2.10, 4.9]{Mackenzie:1992}.

Previous to \cite{Mackenzie:1992}, Lu and Weinstein \cite{LuW:1989} had
shown that associated to any Poisson Lie group $G$ there is a (symplectic)
double groupoid $(S;G,G^*;1)$ which is locally of the form (\ref{diag:vac}).

These results confirm in global form that the structure of double groupoids
is intrinsically associated with Poisson group structures. The deep discovery
by Lu \cite{Lu:1997} of vacant double structures associated to Poisson
actions provides further evidence.

We have so far glossed over the symplectic structure on the double groupoid
of a Poisson Lie group. This has been possible since the Lie algebroid
determined by the Poisson structure on the base of a symplectic groupoid
coincides with the Lie algebroid of the groupoid structure \cite{CosteDW}.
However we now need to utilize these structures.

\section{Duals of double structures}
\label{sect:dds}

It is well known that there is a loose equivalence between the
theories of Lie algebroids and of Poisson structures: a Poisson structure
on a manifold $M$ determines a Lie algebroid structure on $T^*M$
(see \cite{Huebschmann:1990} for history and references), and a Lie
algebroid $A$ gives rise to a (linear) Poisson structure on $A^*$
\cite{Courant:1990}. These processes are nontrivial in the precise sense
that they take place outside the original categories, involving the modules
of sections of the Lie algebroids and the cotangents and tangents (or
rings of functions) of the Poisson manifolds. It is consequently often
valuable to convert problems of Lie algebroid theory into Poisson terms,
and vice versa.

The symplectic groupoid $T^*G\gpd A^*G$, where $G\gpd M$ is any Lie
groupoid, is fundamental to integrability problems in Poisson geometry.
Constructed first by Coste, Dazord and Weinstein \cite{CosteDW}, it was
obtained by Pradines \cite{Pradines:1988} as an instance of a general
duality process applied to the much simpler tangent prolongation groupoid
$TG\gpd TM$. This duality, as we shall see, provides a systematic means of
handling cotangent structures, despite their nonfunctoriality.

The cotangent groupoid provides the simplest important example of
symplectic groupoids, but it is also necessary for the definition of
Poisson groupoid which is most useful here: a Lie groupoid $G$ with a
Poisson structure $\pi^\#\colon T^*G\to TG$ is a Poisson groupoid if
$\pi^\#$ is a groupoid morphism (\cite{AlbertD:1991ssr},
\cite[8.1]{MackenzieX:1994}).

We will see that it is correspondingly necessary, for the understanding of
a double groupoid $(S;H,V;M)$, to place a double groupoid structure on
$T^*S$. Since $S$ has two groupoid structures, $T^*S$ has two cotangent
groupoid structures, over $A^*_HS$ and $A^*_VS$. We must show that these
duals are themselves groupoids over a common base.

Consider a \VBgpd\ $(\Omega;G,A;M)$ as in Figure~\ref{fig:Om}(a), and let
$K$ denote the intersection of the kernels of the source projection
$\Omega\to A$ and the bundle projection $\Omega\to G$. Call $K$ the {\em core}
of $\Omega$; it has a vector bundle structure over $M$ obtained from that
in $\Omega\to G$ \cite[\S4]{Mackenzie:1992}. Pradines \cite{Pradines:1988}
defined a groupoid structure on the vector bundle $\Omega^*$ with base
$K^*$ which makes $(\Omega^*;G,K^*;M)$ a \VBgpd\ with core $A^*$. If moreover
$\Omega$ is an \LAgpd, then the core has a natural induced Lie algebroid
structure \cite[\S4]{Mackenzie:1992}. When $\Omega = TG$ the core is
$AG$ and the dual \VBgpd\ defines the cotangent groupoid structure.

For a double Lie groupoid $S$, Brown and the author \cite{BrownM:1992}
defined the core $C$ to be the intersection of the kernels of the
source projections. There is a canonical groupoid structure on $C$ with
base $M$ which is induced by (but is not a restriction of) the structures
on $S$. In the presence of a double version of local triviality, $C$ and
its associated structure actually determine $S$.

Finally, the \LAgpd s $A_VS$ and $A_HS$ both have core $AC$, and both induce
the usual Lie algebroid structure on $AC$ (\cite[1.6]{Mackenzie:Doubla2}).
It therefore follows that the dual \LAgpd s are
$(A^*_VS;H,A^*C;M)$ and $(A^*_HS;A^*C,V;M)$. The first part of the following
theorem is a long but routine check.

\begin{theorem}
%{\em 
[{\cite[1.4, 1.5]{Mackenzie:SDGDPG}}] %}
With the structures above, $(T^*S;A^*_VS;A^*_HS;A^*C)$
is a double Lie groupoid, and has core groupoid $T^*C\gpd A^*C.$
\end{theorem}

With its canonical symplectic structure, $T^*S$ is the model of a
symplectic double groupoid. More generally, we define a {\em Poisson double
groupoid} to be a double Lie groupoid $({\mathcal S};{\mathcal H}, {\mathcal V};P)$
with a Poisson structure $\pi^\#\colon T^*{\mathcal S}\to T{\mathcal S}$ which is a
morphism of double groupoids from $T^*{\mathcal S}$ to the tangent double
groupoid $(T{\mathcal S};T{\mathcal H},T{\mathcal V};TP)$ \cite{Mackenzie:SDGDPG}.
It then follows that ${\mathcal H}$ and ${\mathcal V}$ are Poisson groupoids, with
respect to the induced structures, both inducing the same Poisson structure
on $P$. Further, taking into account the bundle structures of $T^*{\mathcal S}$
and $T{\mathcal S}$ and treating them as triple structures, the restriction
of $\pi^\#$ to certain corners and cores gives Lie algebroid morphisms
$D_V\colon  A^*{\mathcal H}\to A{\mathcal V}$ and
$D_H\colon  A^*{\mathcal V}\to A{\mathcal H}$ with $D_H^* = -D_V$. When $\pi^\#$
is symplectic these are both isomorphisms of Lie algebroids, and
${\mathcal H}$ and ${\mathcal V}$ are Poisson groupoids in duality in the sense of
Weinstein \cite[\S4.5]{Weinstein:1988}. Taking ${\mathcal S} = T^*S$ gives the
following crucial result.

\begin{theorem}                                   
%{\em 
[{\cite[2.12]{Mackenzie:SDGDPG}}] \label{thm:2.12}
Let $(S;H,V;M)$ be a double Lie groupoid with core $C$.
Then $A^*_VS\gpd A^*C$ and $A^*_HS\gpd A^*C$ are
Poisson groupoids in duality.
\end{theorem}

The duality isomorphisms $A^*(A^*_HS)\to A(A^*_VS)$ and
$A^*(A^*_VS)\to A(A^*_HS)$ can be described in terms which generalize
the description in \cite[7.3]{MackenzieX:1994} of the canonical
symplectic structure on a cotangent bundle in terms of the Tulczyjew
isomorphism $\alpha\colon T^*TM\to TT^*M$ and the map $R$ mentioned in
\S1. See \cite[3.9]{Mackenzie:SDGDPG}.

We have therefore associated to any double Lie groupoid a pair of
dual Lie bialgebroids with base $A^*C$. This is the key to the results
of \S\ref{sect:main}.

\section{Lie bialgebroids and double Lie algebroids}
\label{sect:main}

We suppose given a double vector bundle \cite{Pradines:DVB} as in
Figure~\ref{fig:dvb}; that is, each side has a vector bundle structure,
and the two structures
\begin{figure}[hbt]
\begin{picture}(360,100)
\put(120,40){$\begin{matrix} && \tilq_H           &&\cr
                            &{\mathcal A}&\mlra &A^V&\cr
                            &&&&\cr
                    \tilq_V &\Bigg\downarrow&&\Bigg\downarrow&q_V\cr
                            &&&&\cr
                            &A^H&\mlra &M&\cr
                            &&q_H&&\cr
                            &&&&\cr
\end{matrix}$}
\end{picture}
\caption{\ \label{fig:dvb}}
\end{figure}
on ${\mathcal A}$ {\em commute} in the sense that the maps defining each
structure on ${\mathcal A}$ (the bundle projection, zero section, addition
and scalar multiplication) are morphisms with respect to the other.
We suppose given Lie algebroid structures on each of the four sides, and
want to describe what it means for the two Lie algebroid structures on
${\mathcal A}$ to commute. There are certain obvious requirements, which
need no explanation. To express these efficiently, define an {\em \LAvb}\
to be a double vector bundle as in Figure~\ref{fig:dvb} together with
Lie algebroid structures on a pair of parallel sides, such that the
structure maps of the other pair of vector bundle structures are
Lie algebroid morphisms.

The definition of double Lie algebroid is made up of three conditions.

%\bigskip\noindent{\bf Condition I:} 
\begin{coni}
{\em With respect to the two vertical Lie algebroids,
${\mathcal A}\to A^H$ and $A^V\to M$, the double vector bundle ${\mathcal A}$
is an \LAvb. Likewise, with respect to the two horizontal Lie algebroids,
${\mathcal A}\to A^V$ and $A^H\to M$, the double vector bundle ${\mathcal A}$
is an \LAvb.}
\end{coni}
%\bigskip

Denote the four anchors by $\tila_V\colon {\mathcal A} \to T(A^H),\
\tila_H\colon {\mathcal A}\to T(A^V),\ a_V\colon A^V\to TM$ and
$a_H\colon A^H\to TM$. It is automatic that $\tila_V$ is a morphism of
Lie algebroids over the fixed base $A^H$, and that $a_V$ is a morphism
of Lie algebroids over $M$.

%\bigskip\noindent{\bf Condition II:} 
\begin{coni}
{\em The anchors $\tila_V$ and $a_V$ form a morphism
$(\tila_V, a_V)$
of Lie algebroids with respect to the horizontal structure on ${\mathcal A}$
and the prolongation to $TA^H\to TM$ of the structure on $A^H\to M$.
Likewise, the anchors $\tila_H$ and $a_H$ form a morphism of Lie
algebroids with respect to the vertical structure on ${\mathcal A}$ and
the prolongation to $TA^V\to TM$ of the structure on $A^V\to M$.}
\end{coni}
%\bigskip

We come now to the main condition. The duality for \VBgpd s of
\S\ref{sect:dds} can be applied to either structure on ${\mathcal A}$ and
yields a {\em vertical dual} ${\mathcal A}^{*V}$ and a {\em horizontal dual}
${\mathcal A}^{*H}$, which are themselves double vector bundles as in
Figure~\ref{fig:dvbduals}. Here $K$ is the core of ${\mathcal A}$,
and the cores of ${\mathcal A}^{*V}$ and ${\mathcal A}^{*H}$ are
$(A^V)^*$ and $(A^H)^*$ respectively. Remarkably, ${\mathcal A}^{*V}$
and ${\mathcal A}^{*H}$ are themselves dual.
%
\begin{figure}[hbt]
\begin{picture}(360,100)
\put(20,40){$\begin{matrix}&& \tilq^{(*)}_V &&\cr
        &{\mathcal A}^{*V}&\mlra   &K^*&\cr
        &&&&\cr
\tilq_{*V} &\Bigg\downarrow& &\Bigg\downarrow&q_{K^*} \cr
        &&&&\cr
        &A^H&\mlra &M&\cr
        &&q_H&&\cr
        &&&&\cr
        &&\mbox{(a)}&&
        \end{matrix}$}
%
\put(190,40){$\begin{matrix}&& \tilq_{*H} &&\cr
        &{\mathcal A}^{*H}&\mlra   &A^V&\cr
        &&&&\cr
\tilq^{(*)}_H &\Bigg\downarrow& &\Bigg\downarrow&q_V \cr
        &&&&\cr
        &K^*&\mlra &M&\cr
        &&q_{K^*}&&\cr
        &&&&\cr
        &&\mbox{(b)}&&
	  \end{matrix}$}
\end{picture}
\caption{\ \label{fig:dvbduals}}
\end{figure}

\begin{theorem}
%{\bf 
[{\cite[3.1, 3.5]{Mackenzie:SDGDPG}}] %} 
There is a nondegenerate pairing
of the bundles \break 
${\mathcal A}^{*V}\to K^*$ and ${\mathcal A}^{*H}\to K^*$ given by
\begin{equation}                       \label{eq:3duals}
\langle\Phi, \Psi\rangle = \langle\Psi, \xi\rangle
                            -  \langle\Phi, \xi\rangle
\end{equation}
where $\Phi\in {\mathcal A}^{*V},\ \Psi\in {\mathcal A}^{*H}$ have
$\tilq_V^{(*)}(\Phi) = \tilq_H^{(*)}(\Psi)$ and $\xi$ is any element
of ${\mathcal A}$ with $\tilq_V(\xi) = \tilq_{*V}(\Phi)$ and
$\tilq_H(\xi) = \tilq_{*H}(\Psi).$

The induced isomorphism
$Z_V\colon ({\mathcal A}^{*H})^\dagger\to{\mathcal A}^{*V}$,
where ${}^\dagger$ denotes the dual over $K^*$, is an isomorphism of
double vector bundles over $A^H$ and $K^*$, inducing $-\myid$ on the cores
$(A^V)^*$.
\end{theorem}

This theorem has also been found very recently by Konieczna and Urba\'nski
\cite{KoniecznaU}. Note that the pairing on the LHS of (\ref{eq:3duals}) is
over $K^*$, whereas the pairings on the RHS are over $A^V$ and
$A^H$ respectively. When ${\mathcal A} = TA$ as in Figure~\ref{fig:T2M}(b),
$Z_V = R\circ I^\dagger$ in terms of the maps $R$ and $I$ of
\cite[\S5]{MackenzieX:1994}.

It follows from Condition~I and \cite[3.14]{Mackenzie:SDGDPG} that the
Poisson structure on ${\mathcal A}^{*H}$, which is automatically linear over
$A^V$, is also linear over $K^*$, and therefore induces a Lie algebroid
structure on its dual $({\mathcal A}^{*H})^\dagger$. We use $Z_V$ to transfer
this to ${\mathcal A}^{*V}\to K^*$. Similarly the Poisson structure on
${\mathcal A}^{*V}$ is linear over $K^*$ and therefore induces a Lie algebroid
structure on $({\mathcal A}^{*V})^\dagger\to K^*$ which may be transferred to
${\mathcal A}^{*H}$.

%\bigskip\noindent{\bf Condition III:} 
\begin{coni}
{\em With respect to these structures,
$({\mathcal A}^{*V}, {\mathcal A}^{*H})$ is a Lie bialgebroid over $K^*$.
Further,
$({\mathcal A}^{*V}; AH, K^*; M)$ is an \LAvb\ with respect to the horizontal
Lie algebroid structures and
$({\mathcal A}^{*H}; K^*, AV ;M)$ is an \LAvb\ with respect to the vertical
structures.
}
\end{coni}
\begin{definition}                                    
%{\bf 
[{\cite{Mackenzie:DLADLB}}] \label{df:doubla}
A {\em double Lie algebroid} is a double vector bundle as in
Figure~\ref{fig:dvb} equipped with Lie algebroid structures on all
four sides such that the above Conditions I, II, III are satisfied.
\end{definition}

When $({\mathcal A};A^H, A^V;M)$ is a double Lie algebroid we call
$({\mathcal A}^{*V}, {\mathcal A}^{*H})$ the {\em associated Lie
bialgebroid}.

The notion of Lie bialgebroid was defined by the author and Xu in
\cite{MackenzieX:1994} in terms of the coboundary operators associated to
$A$ and to $A^*$; a more efficient and elegant reformulation was then given
by Kosmann-Schwarzbach in \cite{Kosmann-Schwarzbach:1995}. The criterion
most useful in the current setting is the following.

\begin{theorem}                                        
%{\bf 
[{\cite[6.2]{MackenzieX:1994}}] \label{thm:6.2}
Let $A$ be a Lie algebroid on $M$ such that its dual vector bundle $A^{*}$
also has a Lie algebroid structure. Denote their anchors by $a, a_*$.
Then $(A, A^{*})$ is a Lie bialgebroid if and only if
%$$
\[
T^*(A^*)\buildrel{R}\over\longrightarrow T^*(A)
\buildrel\pi^\#_A\over\longrightarrow TA
%$$
\]
is a Lie algebroid morphism over $a_*$, where the domain
$T^{*}(A^{*})\to A^{*}$ is the cotangent Lie algebroid induced by the
Poisson structure on $A^{*}$, and the target $TA\to TM$ is the tangent
prolongation of $A$.
\end{theorem}

\begin{example}
%{\bf 
[{\cite{Mackenzie:DLADLB}}] %}
Let $(S;H,V;M)$ be a double Lie groupoid. By \ref{thm:2.12}, $A^*_HS$ and
$A^*_VS$, the duals of the \LAgpd s of $S$, are a dual pair of Poisson
groupoids over $A^*C$ with respect to the dual Poisson structures. Further
\cite[\S3]{Mackenzie:SDGDPG}, $A^*(A_HS)$, the vertical dual of $A_2S$,
is canonically isomorphic to $A(A^*_VS)$; this canonical isomorphism
extends the well-known $T^*TM\isom TT^*M$ of \cite{Tulczyjew}. Identifying
$A_2S$ with $A^2S$, the horizontal dual is $A^*(A_VS)\isom A^*(A^*_VS)$,
by a generalization of the map $R$. Thus $({\mathcal A}^{*V}, {\mathcal A}^{*H})$
can be identified with the Lie bialgebroid of $A^*_VS\gpd A^*C$, which
establishes Condition~III.

More generally, dualizing any \LAgpd\ $(\Om;G,A;M)$ gives a Poisson
groupoid $\Om^*\gpd K^*$ \cite[3.14]{Mackenzie:SDGDPG} whose Lie bialgebroid
shows that $(A\Om;AG,A;M)$, as in Figure~\ref{fig:Om}(b), is a double
Lie algebroid.
\end{example}

We come now to the concept of double for a Lie bialgebroid. It was shown
in \cite[2.11]{Mackenzie:SDGDPG} that if $G\gpd P$ is a Poisson groupoid
arising from a symplectic double groupoid $(S;G,G^*;P)$, then the double
Lie algebroid of $S$ is the double cotangent
$T^*(AG)\isom T^*(AG^*)\isom T^*(A^*G)$. For a
Poisson group $G$, such a symplectic double groupoid always exists
\cite{LuW:1989}. In the Poisson groupoid case $G^*$ may not exist (but
see \cite{MackenzieX2}) and it seems unlikely that, even when it does, a
symplectic double groupoid $S$ for $G$ and $G^*$ necessarily exists. We
therefore wish to formulate the double cotangent structure without
assuming that the Lie bialgebroid comes from a Poisson groupoid.

Let $A$ be a Lie algebroid on $M$ such that $A^*$ has
a Lie algebroid structure, not {\em a priori} related to that on $A$.
The structure on $A^*$ induces a Poisson structure on $A$, and
this gives rise to a cotangent Lie algebroid $T^*A\to A$. Equally, the Lie
algebroid structure on $A$ induces a Poisson structure on $A^*$ and this
gives rise to a cotangent Lie algebroid $T^*A^*\to A^*$. We transfer this
latter structure to $T^*A\to A^*$ via $R$.

There are now four Lie algebroid structures on the four sides of
${\mathcal A} = T^*A$ as in Figure~\ref{fig:T*A}.

\begin{theorem}                                       
%{\bf 
[{\cite{Mackenzie:DLADLB}}] \label{thm:Manin}
Let $A$ be a Lie algebroid on $M$ such that its dual vector bundle $A^{*}$
also has a Lie algebroid structure. Then $(A, A^*)$ is a Lie bialgebroid
if and only if ${\mathcal A} = T^*A$, with the structures just described, is a
double Lie algebroid.
\end{theorem}

When $(A,A^*)$ is a Lie bialgebroid, the Lie bialgebroid associated to
its double is effectively $(TA, T(A^*))$, the tangent prolongation over
$TM$. The Manin
triple theorem as given by \cite[\S1]{LuW:1990} characterizes a Lie
bialgebra in terms of the structure of the double on
${\mathfrak g}\oplus{\mathfrak g}^*$ built out of the given structures on
${\mathfrak g}$ and ${\mathfrak g}^*$ and the two coadjoint representations.
In the algebroid case the presence of the core $T^*M$ precludes a
corresponding structure over the base $M$, and we have used instead the
side structures over $A$ and $A^*$. (It is shown in \cite[\S2]{Mackenzie:1992}
that the diagonal structure associated with a matched pair of groupoids,
defined in terms of analogues of the dressing transformation actions,
is a consequence of the triviality of the core; the cotangent double in
Figure~\ref{fig:T*A} has trivial core if and only if the base is trivial.)
By considering the duality (\ref{eq:3duals}) one can show that
a double Lie algebroid as in Figure~\ref{fig:dvb}, for which both
structures on ${\mathcal A}$ are cotangents, must be of the form of
Figure~\ref{fig:T*A}.

The results treated here show that double Lie algebroids stand in the same
relationship to Lie bialgebroids as ordinary Lie algebroids do to
general Poisson manifolds. Any (ordinary) Lie algebroid gives rise to
a Poisson structure on its dual---a Poisson structure which is
necessarily linear---and any Poisson manifold gives rise to a Lie algebroid
structure on its cotangent. These processes are not equivalences of
categories, but it is nonetheless very valuable to be able to pass from one
setting to the other by means of them. It is in the same sense that we say
that Drinfel'd doubles and Ehresmann doubles are coextensive:
Theorem~\ref{thm:Manin} shows that the Drinfel'd double of a
Lie bialgebroid has a double structure in the sense of Ehresmann, and
conversely \ref{thm:2.12} and Condition~III of \ref{df:doubla}
demonstrate that every double Lie groupoid or double Lie
algebroid---where \kwo{double} is now used in the Ehresmann sense---has
hidden in its structure as a key ingredient, a dual pair of Lie bialgebroids
and their Drinfel'd double.

We have concentrated here on the characterization of Lie bialgebroids in
terms of their double cotangent Lie algebroids. However, there are a number
of other characterization results for particular classes of double Lie
algebroids that could be given at this stage. Firstly, it can be shown
\cite{Mackenzie:DLADLB}
that if a double Lie algebroid is {\em vacant}---that is, the core is
the zero bundle---then it defines a matched pair structure
(as in Mokri \cite{Mokri:1997}) on the two side Lie algebroids.
Whereas matched pairs of groups and groupoids can be defined in terms of
conditions on their elements, this characterization is in terms of diagrams
of maps, and may well be capable of extension to other contexts arising
in quantization.

It is also clear that some of the unexpected features of the work of Lu
\cite{Lu:1997} on Poisson homogeneous spaces can be understood in terms of
the dualities studied here.

Lastly, we believe that the calculus developed by Liu, Weinstein and Xu
\cite{LiuWX:1997} will emerge naturally from the study of the double and
triple structures introduced here. This is the subject of ongoing work.

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