LIE BIALGEBROIDS AND POISSON GROUPOIDS
KIRILL C. H. MACKENZIE AND PING XU
1. Introduction
Lie bialgebras arise as infinitesimal invariants of Poisson Lie groups. A Lie
bialgebra is a Lie algebra g with a Lie algebra structure on the dual g∗ which
is compatible with the Lie algebra g in a certain sense. For a Poisson group G,
the multiplicative Poisson structure π induces a Lie algebra structure on the Lie
algebra dual g∗ which makes (g, g∗ ) into a Lie bialgebra. In fact, there is a one-one
correspondence between Poisson Lie groups and Lie bialgebras if the Lie groups
are assumed to be simply connected [7], [16], [19]. The importance of Poisson
Lie groups themselves arises in part from their role as classical limits of quantum
groups [8] and in part because they provide a class of Poisson structures for which
the realization problem is tractable [15].
Poisson groupoids were introduced by Weinstein [24] as a generalization of both
Poisson Lie groups and the symplectic groupoids which arise in the integration of
arbitrary Poisson manifolds [4], [11]. He noted that the Lie algebroid dual A∗ G
of a Poisson groupoid G itself has a Lie algebroid structure, but did not develop
the infinitesimal structure further. In this paper we introduce and study a natural
infinitesimal invariant for Poisson groupoids, the Lie bialgebroids of the title. Our
ultimate purpose is to develop a Lie theory for Poisson groupoids parallel to that
for Poisson groups. In this paper we are primarily concerned with the first half
of this process; that is, with the construction of the Lie bialgebroid of a Poisson
groupoid.
After the preliminary §2, in which we describe the generalization to arbitrary
Lie algebroids of the exterior calculus and Schouten calculus, in §3 we define a
Lie bialgebroid to be a Lie algebroid A whose dual A∗ is also equipped with a Lie
algebroid structure, such that the coboundary operator d∗ : A −→ ∧2 (A) associated
to A∗ satisfies a cocycle equation with respect to Γ(A), the Lie algebra of sections
of A. This is clearly a straightforward extension of the concept of a Lie bialgebra
[16] but cannot be formalized in Lie algebroid cohomological terms since there is
no satisfactory adjoint representation for a general Lie algebroid. Most of §3 is
devoted to proving that this definition is self-dual: if (A, A∗ ) is a Lie bialgebroid,
then (A∗ , A) is also. In §4, we briefly consider the special case of Lie bialgebroids
satisfying a triangularity condition, which include some important examples such
as the usual triangular Lie bialgebras and Lie bialgebroids associated to Poisson
manifolds.
The techniques used in §§2–4 are similar to those known for Lie bialgebras. It
would be possible, by suitably generalizing the proof for Poisson groups, to prove
1991 Mathematics Subject Classification. Primary 58F05. Secondary 17B66, 22A22, 58H05.
Research of the second author partially supported by NSF Grant DMS92-03398, and his research at MSRI supported by NSF Grant DMS90-22140.
1
2
K. C. H. MACKENZIE AND PING XU
already at this stage that if G is a Poisson groupoid, then (AG, A∗ G) is a Lie
bialgebroid with respect to the canonical structures. However we prefer to deduce
this in §8 from the results of §6 and §7.
The integrability of Lie bialgebras is established by noting that for a Poisson
Lie group G, the Poisson tensor can be right-translated to the identity element and
gives a Lie group 1-cocycle on G with values in g∗ . Since every finite-dimensional
Lie algebra is integrable to a simply-connected Lie group, the integrability problem
for Lie bialgebras is equivalent to the problem of lifting Lie algebra cocycles to Lie
group cocycles, and this is always possible when the Lie group is simply-connected.
In the case of a Lie bialgebroid (A, A∗ ), there is first of all the fact that not all
Lie algebroids A integrate to Lie groupoids; in discussing the integrability of Lie
bialgebroids we will however always assume that the Lie algebroids themselves are
integrable. More importantly, what needs to be lifted is now a cocycle of an infinitedimensional Lie algebra Γ(A), and here there is no general theory to apply. Instead
we propose to make use of the machinery developed for Lie algebroid morphisms
in [9]. To explain how this works, we have to recall briefly the relation between
Poisson Lie groups and Lie bialgebras from this viewpoint.
In [18, §4] one of us analysed the relations between a general Poisson Lie group
and its Lie algebra dual in terms of groupoids and Lie algebroids. In particular, it
is shown that a Lie group with a Poisson structure is a Poisson Lie group if and
only if the Poisson bundle map from T ∗ G to T G is a Lie groupoid morphism, where
T ∗ G is the canonical cotangent groupoid over the base g∗ and T G is the tangent
group.
This result can be easily generalized to the case of groupoids in a straightforward
manner (§8). This suggests that there should be an equivalent description of Lie
bialgebroids in terms of Lie algebroid morphisms, and this is in fact the content
of §6. Such a description appears to be more complicated than Definition 3.1, but
should be more tractable when dealing with the integrability problem. We prove in
Theorem 6.2 that a Lie algebroid A with a Lie algebroid structure on its dual A∗
#
: T ∗ (A) −→ T A is a
is a Lie bialgebroid if and only if the Poisson bundle map πA
Lie algebroid morphism with respect to the Lie algebroid structure on T ∗ (A) with
base A∗ . This result, the proof of which extends over all of §6, is the main result of
the paper, and is the key to the relationship between Lie bialgebroids and Poisson
groupoids. In order to formulate and prove Theorem 6.2, we need to develop the
basic properties of the tangent and cotangent bundles of a Lie algebroid, and we
do this in §5. In particular, we show that the tangent T A −→ T P of any Lie
algebroid A −→ P is itself a Lie algebroid in a natural way; this structure is dual
to the tangent Poisson structure [6] for the dual Poisson structure on A∗ . Similarly,
before we can deal with Poisson groupoids we need to explicate the double structures
associated with the tangent and cotangent bundles of a Lie groupoid, and we do
this in §7. Sections 5 and 7 may be regarded as a direct continuation of some
aspects of [18].
Finally, we note that given a Lie bialgebra (g, g∗ ) it is a simple matter to verify
that g ⊕ g∗ has a Lie algebra structure for which g and g∗ are subalgebras, and for
which the cross terms are given by the two coadjoint representations. This is the so
called double Lie algebra of (g, g∗ ). The importance of the double Lie algebra lies
in the fact that it embodies almost all the information of the Lie bialgebra, and its
corresponding Lie group, the so called double Lie group, turns out to be a natural
LIE BIALGEBROIDS AND POISSON GROUPOIDS
3
object describing the dressing transformations. For a Lie bialgebroid (A, A∗ ) on an
arbitrary base, the usual formula does not give a Lie algebroid structure on A ⊕ A∗ .
In fact, evidence suggests that the correct replacement should be T ∗ (A). However,
the structure on T ∗ (A) which will play the role of the double Lie algebra is not
clear, and we hope to explore this in the future.
Acknowledgement This work was initiated when the second author visited the
University of Sheffield in the summer of 1991. He wishes to thank the University
of Sheffield for its hospitality during the visit.
Notation For the convenience of the reader, we give a list of the most common
notations used throughout the paper.
q : A −→ P
q∗ : A∗ −→ P
pP : T P −→ P
cP : T ∗ P −→ P
m, n
x, y, z
ω, θ
X, Y, Z
φ, ψ
ξ, η
Φ, Ψ
X, Y
F, S
f, s
I : T (A∗ ) −→ T • (A)
R : T ∗ (A∗ ) −→ T ∗ (A)
vector bundle or Lie algebroid
dual vector bundle or Lie algebroid
tangent bundle of manifold P
cotangent bundle of manifold P
elements of P
tangent vectors or vector fields on P
cotangent vectors or 1-forms on P
elements or sections of A
elements or sections of A∗
tangent vectors or vector fields on A
cotangent vectors or 1-forms on A
tangent vectors or vector fields on A∗
cotangent vectors or 1-forms on A∗
elements or sections of the dual of T A −→ T P
canonical isomorphism (5.3)
canonical isomorphism (5.5)
2. Differential calculus on Lie algebroids
As a preliminary, in this section we describe the generalization to arbitrary Lie
algebroids of the calculuses of differential forms and multivector fields.
Consider a Lie algebroid A on base P with anchor a : A −→ T P . The generalization to A of the standard calculus of differential forms has been treated at
length in [17, IV §2] and references given there. Here we give only a quick summary. For k ≥ 0, let ∧k (A∗ ) denote the kth exterior power bundle on P ; we identify
∧k (A∗ ) with Altk (A, P × R). The exterior derivative, or Lie algebroid coboundary,
d : Γ(∧k A∗ ) −→ Γ(∧k+1 A∗ ) is defined by
dφ(X1 , . . . , Xk+1 )
=
k+1
X
(−1)i+1 a(Xi )(φ(X1 , . ˆ. ., Xk+1 ))
i=1
+
X
i <j
(−1)i+j φ([Xi , Xj ], X1 , . ˆ. . . ˆ. . , Xk+1 ),
4
K. C. H. MACKENZIE AND PING XU
for φ ∈ Γ(∧k A∗ ), Xi ∈ ΓA, 1 ≤ i ≤ k + 1. For X ∈ ΓA and k ≥ 0, the Lie
derivative LX : Γ(∧k A∗ ) −→ Γ(∧k A∗ ) is defined by
LX (φ)(Y1 , . . . , Yk ) = a(X)(φ(Y1 , . . . , Yk )) −
k
X
φ(Y1 , . . . , [X, Yi ], . . . , Yk ),
(1)
i=1
for φ ∈ Γ(∧k A∗ ), Y1 , . . . , Yk ∈ ΓA. The contraction, or interior multiplication,
ιX : Γ(∧k+1 A∗ ) −→ Γ(∧k A∗ ) is defined by
ιX (φ)(Y1 , . . . , Yk ) = φ(X, Y1 , . . . , Yk ),
k+1
∗
for φ ∈ Γ(∧
A ), Y1 , . . . , Yk ∈ ΓA.
These operators satisfy the following analogues of the standard calculus of differential forms; see [17, IV §2].
d(φ ∧ ψ)
d
2
L[X,Y ]
ι[X,Y ]
LX
Lf X (φ)
∞
= dφ ∧ ψ + (−1)k φ ∧ dψ
=
(2)
0
(3)
= LX ◦ LY − LY ◦ LX
(4)
= LX ◦ ιY − ιY ◦ LX
(5)
= d ◦ ιX + ιX ◦ d
(6)
= f LX (φ) + df
k
∗
∧ ιX (φ)
m
(7)
∗
where X, Y ∈ ΓA, f ∈ C (P ), φ ∈ Γ(∧ A ), ψ ∈ Γ(∧ A ). Note that in (7),
df ∈ Γ(∧1 A∗ ) refers to the Lie algebroid coboundary.
In a similar way the Schouten bracket and Lie derivative of multivector fields
extend to A. We refer to a section D ∈ Γ(∧k A) as a k-multisection of A. The
generalized Schouten bracket
[ , ] : Γ(∧k A) × Γ(∧m A) −→ Γ(∧k+m−1 A)
is characterized by the conditions that [ , ] : Γ(∧1 A) × Γ(∧1 A) −→ Γ(∧1 A) coincide
with the Lie algebroid bracket, that [X, f ] = a(X)(f ) for X ∈ ΓA, f ∈ C ∞ (P ),
and that the properties
[D, D0 ] = −(−1)(k−1)(m−1) [D0 , D],
(8)
(−1)(k−1)(n−1) [[D, D0 ], D00 ] + (−1)(m−1)(k−1) [[D0 , D00 ], D]
+ (−1)(n−1)(m−1) [[D00 , D], D0 ] = 0,
[D, D0 ∧ D00 ] = [D, D0 ] ∧ D00 + (−1)k(n−1) D0 ∧ [D, D00 ],
(9)
(10)
hold for all D ∈ Γ(∧k A), D0 ∈ Γ(∧m A), D00 ∈ Γ(∧n A). (Compare [12], [13].)
We denote the pairing between a k-form φ and a k-multisection D of A variously
by φ · D or hD, φi, according to whichever is clearest in the given context. Given
φ ∈ Γ(∧k A∗ ), there is a contraction, ιφ : Γ(∧m A) −→ Γ(∧m−k A), where m ≥ k,
defined by
(ιφ (D))(ψ) = (φ ∧ ψ) · D.
For X ∈ ΓA and D ∈ Γ(∧k A) we also write
LX (D) = [X, D];
LIE BIALGEBROIDS AND POISSON GROUPOIDS
5
this is the Lie derivative of multisections. It follows from (8)—(10) above that
LX : Γ(∧k A) −→ Γ(∧k A) has the following properties:
LX (D ∧ D0 )
=
LX (D) ∧ D0 + D ∧ LX (D0 )
(11)
[LX , LY ]
=
L[X,Y ]
(12)
LX (f D)
=
f LX (D) + a(X)(f )D
(13)
Lf X (D)
=
f LX (D) − X
(14)
∧ ιdf (D)
0
where X, Y ∈ ΓA, D ∈ Γ(∧k A), D0 ∈ Γ(∧k A), f ∈ C ∞ (P ).
Finally, for φ ∈ Γ(∧k A∗ ), D ∈ Γ(∧k A), X ∈ ΓA, we have
LX (φ · D) = LX (φ) · D + φ · LX (D).
(15)
Pk
This follows from (1) and LX (Y1 ∧ . . . ∧ Yk ) = i=1 Y1 ∧ . . . ∧ [X, Yi ] ∧ . . . ∧ Yk .
Example 2.1. Let P be a Poisson manifold with Poisson tensor π. It is well-known
that the cotangent bundle T ∗ P −→ P carries a natural Lie algebroid structure [4],
[5]. Given f ∈ C ∞ (P ), denote the Hamiltonian vector field corresponding to f by
Xf . Then the anchor π # : T ∗ P −→ T P is determined by π # (f δg) = f Xg . Given
ω, θ ∈ Ω1 (P ), and writing Xω = π # ω and Xθ = π # θ, the Lie algebroid bracket is
{ω, θ} = LXω θ − LXθ ω − δ(π(ω, θ)).
With respect to this structure, the Lie derivatives with respect to ω ∈ Ω1 (P ) of a
function f ∈ C ∞ (P ), a 1-form θ, and a vector field X ∈ ΓT P , are given by
Lω (f ) = Xω (f ),
Lω (θ) = {ω, θ},
(Lω X) · θ = Xω (X · θ) − X · {ω, θ}.
Furthermore, the exterior derivative d : Γ(∧k T P ) −→ Γ(∧k+1 T P ) is exactly the
differential operator of the Poisson complex [14], namely d(D) = [π, D], where the
bracket is the Schouten bracket in T ∗ P .
3. Lie bialgebroids
In this section, we introduce the concept of Lie bialgebroid, and develop its
elementary properties. The definition which follows uses the bracket on ΓA and
the coboundary d∗ : Γ(∧k A) −→ Γ(∧k+1 A) associated to A∗ ; we will see at the end
of the section that this is equivalent to the corresponding condition with the roles
of A and A∗ reversed.
Definition 3.1. Suppose that A −→ P is a Lie algebroid, and that its dual bundle
A∗ −→ P also carries a Lie algebroid structure. Then (A, A∗ ) is a Lie bialgebroid
if for any X, Y ∈ Γ(A),
d∗ [X, Y ] = LX d∗ Y − LY d∗ X.
(16)
Remark 3.2. It is clear that Equation (16) is equivalent to requiring that the
coboundary d∗ : Γ(A) −→ Γ(∧2 A) be a Lie algebra 1-cocycle for the infinite dimensional Lie algebra Γ(A), with the module structure on Γ(∧2 A) being the natural
one. However, since d∗ is not C ∞ (P )-linear, it is not a Lie algebroid cocycle in the
sense of [17].
It is obvious that a Lie bialgebra (g, g∗ ) in the sense of Drinfel’d [8] is a special
case of a Lie bialgebroid. Another interesting example is the following.
6
K. C. H. MACKENZIE AND PING XU
Example 3.3. Suppose that P is a Poisson manifold. Let A be the usual tangent
bundle Lie algebroid T P −→ P , and let A∗ = T ∗ P −→ P be equipped with the
canonical Lie algebroid structure induced from the Poisson struture on P . Clearly
(T P, T ∗ P ) is a Lie bialgebroid. In fact, the compatibility condition (16) follows
from the graded Jacobi identity of the Schouten brackets [12].
In what follows, we always use a and a∗ to denote the anchor maps of A and A∗
respectively.
Proposition 3.4. Assume that (A, A∗ ) is a Lie bialgebroid. Then, for any X ∈
Γ(A) and f ∈ C ∞ (P ),
Ldf X = −[d∗ f, X].
Proof. For any X, Y ∈ Γ(A) and f ∈ C ∞ (P ) we have, by Equation (16), that
d∗ [X, f Y ]
= d∗ (f [X, Y ]) + d∗ ((a(X)f )Y )
= d∗ f
∧
[X, Y ] + f (−LY d∗ X + LX d∗ Y ) + d∗ (a(X)f ) ∧ Y
+ (a(X)f )d∗ Y.
On the other hand,
d∗ [X, f Y ]
= −Lf Y d∗ X + LX d∗ (f Y )
= −Lf Y d∗ X + (LX d∗ f ) ∧ Y + (d∗ f ) ∧ LX Y + f LX d∗ Y
+ (a(X)f )d∗ Y.
It thus follows that
−Lf Y d∗ X + (LX d∗ f ) ∧ Y = f (−LY d∗ X) + d∗ (a(X)f ) ∧ Y.
According to Equation (14),
Lf Y d∗ X = f LY d∗ X − Y
∧ ιdf d∗ X,
and so
(LX (d∗ f )) ∧ Y = [ιdf d∗ X + d∗ (a(X)f )] ∧ Y,
which implies, using (6), that
Ldf X = −[d∗ f, X].
Corollary 3.5. Let πP# denote the composition a ◦ a∗∗ : T ∗ P −→ T P . Then we
have
[d∗ g, d∗ f ] = d∗ (πP# (δg)f ).
(17)
Proof. By letting X = d∗ g in Proposition 3.4, it follows that
[d∗ g, d∗ f ] = Ldf d∗ g = d∗ [ιdf (d∗ g)] = d∗ ((a∗ δf ) · (a∗∗ δg)) = d∗ (πP# (δg)f ).
Proposition 3.6. Suppose that (A, A∗ ) is a Lie bialgebroid. Then πP# : T ∗ P −→
T P defines a Poisson structure on P , and so also does π̄P# = a∗ ◦ a∗ : T ∗ P −→ T P .
Moreover, πP# and π̄P# are opposite to one another.
LIE BIALGEBROIDS AND POISSON GROUPOIDS
7
Proof. As a first step, we need to prove that πP# is skew-symmetric. It follows
from Corollary 3.5 that d∗ ((πP# δf )f ) = 0 for any f ∈ C ∞ (P ). In particular,
d∗ ((πP# δf 2 )f 2 ) = 0. It follows immediately that
(πP# (δf )f )d∗ f 2 = 0.
(18)
Applying a to both sides of Equation (18), one obtains that 2f (πP# (δf )f )2 =
0, which yields that πP# (δf )f = 0. This implies immediately that πP# is skewsymmetric, since πP# is a bundle map. Let {g, f } = (πP# δg)f as usual. Applying a
to both sides of Equation (17), we have
πP# δ({g, f }) = [πP# δg, πP# δf ],
which is equivalent to the Jacobi identity of the bracket. So πP# defines a Poisson
structure on P . The rest of the theorem follows immediately.
To avoid confusion in the sequel, by the Poisson structure induced on P by a Lie
bialgebroid (A, A∗ ), we always mean the structure defined by πP# , unless explicitly
stated otherwise. It follows immediately from Corollary 3.5 that a∗∗ : T ∗ P −→ A is
bracket-preserving and a Lie algebroid morphism. In terms of the tangent Poisson
structures of [6], we have the following
Corollary 3.7. Let (A, A∗ ) be a Lie bialgebroid, and let T P be equipped with the
tangent Poisson structure. Then the anchor a∗ : A∗ −→ T P is a Poisson map,
while a : A −→ T P is anti-Poisson.
The next result is dual to, and follows from, Proposition 3.4.
Proposition 3.8. Assume that (A, A∗ ) is a Lie bialgebroid. For any φ ∈ Γ(A∗ )
and f ∈ C ∞ (P ), we have:
Ld∗ f φ = −[df, φ].
Proof. For any X ∈ Γ(A), using (15) twice,
(Ld∗ f φ) · X
= Ld∗ f (φ · X) − φ · (Ld∗ f X)
=
((a ◦ a∗∗ )(δf ))(φ · X) − φ · (Ld∗ f X)
= −((a∗ ◦ a∗ )(δf ))(φ · X) − φ · (Ld∗ f X)
= −Ldf (φ · X) − φ · (Ld∗ f X)
= −[df, φ] · X − φ · (Ldf X + [d∗ f, X])
= −[df, φ] · X.
In the third equality we used the fact that the bundle map a ◦ a∗∗ : T ∗ P −→ T P is
skew-symmetric.
Corollary 3.9. Let (A, A∗ ) be a Lie bialgebroid. Let X ∈ Γ(A), φ ∈ Γ(A∗ ), and
ω ∈ Ω1 (P ). Then
La∗ ω X = −[a∗∗ ω, X] − a∗∗ (ιa(X) δω),
and
La∗∗ ω φ = −[a∗ ω, φ] − a∗ (ιa∗ (φ) δω).
8
K. C. H. MACKENZIE AND PING XU
Proof. We shall only prove the first identity. The second one can be shown similarly.
Without loss of generality, we assume that ω = gδf for some f, g ∈ C ∞ (P ).
Then the left hand side is
Lgdf X = gLdf X + d∗ g ∧ (ιdf X) = −g[d∗ f, X] + (a(X)f )d∗ g,
using Proposition 3.4 and (7), while the right hand side is
−[a∗∗ gδf, X] − a∗∗ (ιa(X) (δg ∧ δf )) = (a(X)f )d∗ g − g[d∗ f, X].
We are now ready to prove the following duality theorem for Lie bialgebroids.
Theorem 3.10. If (A, A∗ ) is a Lie bialgebroid, then so is (A∗ , A).
To prove this theorem, we need the following lemma:
Lemma 3.11. For any X ∈ Γ(A), φ ∈ Γ(A∗ ) and f ∈ C ∞ (P ),
LLφ X f = Lφ LX f − X · [φ, df ].
Proof. This is a simple calculation using the fact that LZ (f ) = df ·Z for all Z ∈ ΓA:
LLφ X f = (Lφ X) · df = Lφ (X · df ) − X · Lφ (df ) = Lφ LX f − X · [φ, df ].
Proof of Theorem 3.10. Let
K = (LX d∗ Y − LY d∗ X − d∗ [X, Y ])(φ ∧ ψ) − (Lφ dψ − Lψ dφ − d[φ, ψ])(X ∧ Y ).
To prove the theorem, it suffices to show that K = 0. It follows from a straightforward computation that
K
= LLψ X (φ · Y ) − LLψ Y (φ · X) − Lψ [LX (φ · Y ) − LY (φ · X)]
− LLφ X (ψ · Y ) + LLφ Y (ψ · X) + Lφ [LX (ψ · Y ) − LY (ψ · X)]
− LLY φ (ψ · X) + LLY ψ (φ · X) + LY [Lφ (ψ · X) − Lψ (φ · X)]
+ LLX φ (ψ · Y ) − LLX ψ (φ · Y ) − LX [Lφ (ψ · Y ) − Lψ (φ · Y )]
=
−X · [ψ, d(φ · Y )] + Y · [ψ, d(φ · X)] + X · [φ, d(ψ · Y )] − Y · [φ, d(ψ · X)]
+ φ · [Y, d∗ (ψ · X)] − ψ · [Y, d∗ (φ · X)] − φ · [X, d∗ (ψ · Y )]
+ ψ · [X, d∗ (φ · Y )]
=
Ld(φ·Y ) (ψ · X) − Ld(φ·X) (ψ · Y ) − Ld(ψ·Y ) (φ · X) + Ld(ψ·X) (φ · Y )
=
0.
Here, in the second step, we have used Lemma 3.11 and its dual, while the third step
depends on Propositions 3.4 and 3.8 in the forms Ldf (X) = −Ld∗ f (X), Ldf (φ) =
−Ld∗ f (φ). The last step uses Ldf (g) = −Ldg (f ), which follows from Proposition
3.6.
LIE BIALGEBROIDS AND POISSON GROUPOIDS
9
4. Triangular Lie bialgebroids
In this section we briefly consider the special case of Lie bialgebroids defined by
a suitable bundle map A∗ −→ A. This includes the case of triangular Lie bialgebras
and the Lie bialgebroids associated to Poisson manifolds. Throughout this section
we let A −→ P be a Lie algebroid with anchor a, and let Λ ∈ Γ(∧2 A) be a bi-section
of A satisfying [Λ, Λ] = 0. Let Λ# : A∗ −→ A denote the bundle map associated to
Λ, and let a∗ = a ◦ Λ# : A∗ −→ T P . Define a bracket of sections of A∗ by
[φ, ψ] = LΛ# φ ψ − LΛ# ψ φ − d(Λ(φ, ψ)) = ιΛ# φ (dψ) − ιΛ# ψ (dφ) + d(Λ(φ, ψ))
(19)
for any φ, ψ ∈ Γ(A∗ ).
Theorem 4.1. The vector bundle A∗ −→ P together with the bracket above and
the anchor a∗ is a Lie algebroid. Moreover, (A, A∗ ) is a Lie bialgebroid.
Proof. For any φ, ψ ∈ Γ(A∗ ), and f ∈ C ∞ (P ),
[φ, f ψ]
=
LΛ# φ f ψ − LΛ# (f ψ) φ − d(Λ(φ, f ψ))
=
f (LΛ# φ ψ − LΛ# ψ φ − d(Λ(φ, ψ))) + (LΛ# φ f )ψ
=
f [φ, ψ] + (a(Λ# φ)(f ))ψ
=
f [φ, ψ] + (a∗ (φ)f )ψ.
The Jacobi identity of the bracket can be checked directly (compare [4], [11]).
Therefore A∗ is a Lie algebroid.
The compatiblity between A and A∗ follows from the graded Jacobi identity of
the Schouten brackets on Γ(∧∗ A) and the following lemma.
Lemma 4.2. For any X ∈ Γ(A),
d∗ X = [Λ, X].
∗
Proof. For any φ, ψ ∈ Γ(A ), we have
(d∗ X)(φ, ψ)
= a∗ (φ)(X · ψ) − a∗ (ψ)(X · φ) − X · [φ, ψ]
= a(Λ# φ)(X · ψ) − a(Λ# ψ)(X · φ)
− X · (LΛ# φ ψ − LΛ# ψ φ − d(Λ(φ, ψ)))
= LΛ# φ (X · ψ) − LΛ# ψ (X · φ) − X · LΛ# φ ψ + X · LΛ# ψ φ
+ X · d(Λ(φ, ψ))
=
(LΛ# φ X) · ψ − (LΛ# ψ X) · φ + a(X)(Λ(φ, ψ))
= −LX (Λ# φ) · ψ + LX (Λ# ψ) · φ + LX (Λ(φ, ψ)
=
Λ(φ, LX ψ) − Λ(ψ, LX φ) + LX (Λ(ψ, φ))
= −(LX Λ)(φ, ψ)
=
[Λ, X](φ, ψ).
We shall call such a Lie bialgebroid, constructed as above, a triangular Lie
bialgebroid. Clearly, triangular Lie bialgebras and the Lie bialgebroids associated to
Poisson manifolds as in Example 3.3 are special cases of triangular Lie bialgebroids.
An important property of a triangular Lie bialgebroid is the following (compare
a similar result for triangular Lie bialgebras [22], [25]):
10
K. C. H. MACKENZIE AND PING XU
Theorem 4.3. Let (A, A∗ ) be a triangular Lie bialgebroid. Then Λ# : A∗ −→ A
is a Lie algebroid morphism.
Proof. Since Λ# is skew-symmetric, for any φ, ψ, γ ∈ Γ(A∗ ) we have
hΛ# {φ, ψ}, γi = −hLΛ# φ ψ, Λ# γi + hLΛ# ψ φ, Λ# γi + hd(Λ(φ, ψ)), Λ# γi
= −LΛ# φ (ψ · Λ# γ) + ψ · [Λ# φ, Λ# γ] + LΛ# ψ (φ · Λ# γ)
− φ · [Λ# ψ, Λ# γ] + [a(Λ# γ)](Λ(φ, ψ))
=
[Λ, Λ](φ, ψ, γ) + γ · [Λ# φ, Λ# ψ]
= h[Λ# φ, Λ# ψ], γi.
Therefore Λ# is a Lie algebroid morphism.
That (19) defines a Lie algebroid structure on A∗ if [Λ, Λ] = 0 was also shown
by Kosmann-Schwarzbach and Magri [12, §6.5].
5. Tangent Lie algebroids and their duals
Lie bialgebras first arose as infinitesimal invariants of Poisson Lie groups. In
order to study the corresponding relationship between Poisson groupoids and Lie
bialgebroids, we need to work extensively with the tangents of vector bundles and
Lie algebroids. There are two main reasons for this. Firstly, whereas the (co)tangent
bundle of a Lie group is trivializable, this is not usually true of Lie groupoids, and
in place of Lie bialgebras of the form g ⊕ g∗ we need to consider the cotangent
bundles of general Lie algebroids. Secondly, the nonlinearity referred to in the
remark following (3.1) can be removed by lifting to the tangent bundle. In this
section, we set out the basic facts which we need about tangent vector bundles and
their duals.
We refer to §1 of [18], and references given there, for background on double vector
bundles, and in particular for the concepts of core and of core sequences. We will
be mostly concerned with the tangent double vector bundle of a vector bundle, and
various double vector bundles associated to it.
Consider a vector bundle q : A −→ P . The tangent double vector bundle
TA
pA
T (q)
−−−−−−−−−−−→




y




y
A
TP
−−−−−−−−−−−→
q
p
(20)
P
is described in [3], [20], and [18], and we briefly recall the structure here. In A and
T P , we use standard notation. The zero of A over m ∈ P is 0A
m , and the zero of
T P over m is 0Tm .
We denote elements of T A by ξ, η, ζ . . . , and we write (ξ; X, x; m) to indicate
that X = pA (ξ), x = T (q)(ξ), and m = p(T (q)(ξ)) = q(pA (ξ)). With respect to the
tangent bundle structure (T A, pA , A), we use standard notation: + for addition,
− for subtraction and juxtaposition for scalar multiplication. The notation TX (A)
will always denote the fibre p−1
A (X), for X ∈ A, with respect to this bundle. The
LIE BIALGEBROIDS AND POISSON GROUPOIDS
11
zero element in TX (A) is denoted e
0X . We refer to this bundle structure as the
pA -bundle structure.
With respect to the T (q)-bundle structure, (T A, T (q), T P ), we use +
+ for addition,
for subtraction, and for scalar multiplication. This addition and scalar
multiplication on T A are precisely the tangents of the addition and scalar multiplication in A. The fibre over x ∈ T P will always be denoted T (q)−1 (x), and the
zero element of this fibre is T (0)(x). If we consider elements ξ of T A as derivatives
of paths in A and write
d Xt ,
ξ=
dt
0
where Xt denotes a path
in
A
defined
in
a
neighbourhood of 0 ∈ R, then pA (ξ) = X0
d
and T (q)(ξ) = dt
q(Xt )0 . If ξ, η ∈ T A have T (q)(ξ) = T (q)(η), then we can arrange
d
d
Yt 0 , where q(Xt ) = q(Yt ) for all t in a neighbourhood
that ξ = dt Xt 0 and η = dt
of 0 ∈ R and then
d
d
ξ+
+ η=
(Xt + Yt ) , λ ξ =
λXt .
dt
dt
.
.
0
0
For each m ∈ P , the tangent space T0m (Am ) identifies canonically with Am ; we
denote the element of T0m (Am ) corresponding to X ∈ Am by X. This identifies A
with the core of T A. Note that, for X, Y ∈ Am and t ∈ R,
.
X +Y =X +Y =X +
+ Y,
tX = tX = t X.
0
0
Given a morphism of vector bundles φ : A −→ A, f : P −→ P , we denote the
pullback of A across f by f ! A, and the induced morphism A0 −→ f ! A over P 0
by φ! . Associated to the double vector bundle structure on T A are the two core
sequences:
τ
T (q)!
q ! A >−−−> T A −−− q ! T P,
(21)
where the vector bundles have base A, and
υ
p!A
p! A >−−−> T A −−− p! A,
(22)
where the vector bundles have base T P . Here τ is the map
τ (X, Y ) = e
0X +
+ Y,
which assigns to (X, Y ) ∈ Am × Am the element of TX (Am ) which has its tail at
X, and is parallel to Y , and υ is the map
υ(x, Y ) = T (0)(x) + Y .
We refer to (21) as the core sequence for pA , and to (22) as the core sequence for
T (q). We call τ and υ translation maps.
Given X ∈ Γ(A) define a vector field X̆ on A by X̆(Y ) = τ (Y, X(qY )), Y ∈ A.
Then
d
F (Y + tX(qY ))
X̆(F )(Y ) =
dt
0
for F ∈ C ∞ (A), Y ∈ A, and so
X̆(f ◦ q) = 0,
X̆(lφ ) = hφ, Xi ◦ q,
[X̆, Y̆ ] = 0,
(23)
for f ∈ C ∞ (P ), X, Y ∈ Γ(A), φ ∈ Γ(A∗ ). Here lφ ∈ C ∞ (A) is the function
X 7→ hφ, Xi. Note also that (f X)˘= (f ◦ q)X̆.
12
K. C. H. MACKENZIE AND PING XU
b of T (q) by X(x)
b
A section X ∈ Γ(A) also induces a section X
= υ(x, X(px)).
Note that
b+
b
(X + Y ) b = X
+ Yb ,
(f X) b = (f ◦ p) X,
.
∞
for X, Y ∈ Γ(A), f ∈ C (P ).
If A is a trivial vector bundle P × V −→ P , where V is a vector space, we denote
elements of
ξ = (x, X, X 0 ), where if mt is a path in P with
T A = T P × V × V by
d
d
0
x = dt mt 0 , we identify (x, X, X ) with dt
(mt , X + tX 0 )0 . The operations in the
bundle T (q) : T P × V × V −→ T P, (x, X, X 0 ) 7→ x, are then given by
(x, X, X 0 ) +
+ (x, Y, Y 0 )
=
(x, X + Y, X 0 + Y 0 ),
t (x, X, X 0 )
=
(x, tX, tX 0 ),
T (0)(x)
=
(x, 0, 0),
.
(24)
where x ∈ T P, X, X 0 , Y, Y 0 ∈ V, t ∈ R. The operations in the tangent bundle
pP ×V : T P × V × V −→ P × V, (x, X, X 0 ) 7→ (px, X), are of course given by
(x, X, X 0 ) + (y, X, Y 0 )
(x + y, X, X 0 + Y 0 ),
=
t(x, X, X 0 ) = (tx, X, tX 0 ),
e
0(m,X) = (0Tm , X, 0),
(25)
where X, X 0 , Y, Y 0 ∈ V, t ∈ R and x, y ∈ T P have p(x) = p(y). Given (m, X) ∈
P × V , the corresponding core element of T P × V × V is
(m, X) = (0Tm , 0, X).
Consider now the case of the double tangent bundle T 2 P , where A = T P . Given
a function f on P we define a function fe on T P by fe(X) = X(f ), and given a
e on T P by X
e = J ◦ T (X), where J
vector field X on P we define a vector field X
2
is the canonical involution of T P . The following identities are easily established:
e + Ye ,
^
X
+Y =X
e ◦ p) = X(f ) ◦ p,
X(f
^
e Ye ],
[X,
Y ] = [X,
e + feX̆,
fg
X = (f ◦ p)X
]),
e fe) = X(f
X(
(26)
e Y̆ ] = [X, Y ]˘,
[X,
where X, Y ∈ X (P ), f ∈ C ∞ (P ).
Now suppose that A is a Lie algebroid on P , with anchor a : A −→ T P . Define
aT = J ◦ T (a) : T A −→ T 2 P ; then aT is a morphism of vector bundles over T P .
Define a bracket on ΓT P T A by the conditions
[T (X), T (Y )] = T ([X, Y ]),
[T (X), Yb ] = [X, Y ] b ,
b Yb ] = 0,
[X,
(27)
for X, Y ∈ Γ(A), and extending over C ∞ (A) by the Lie algebroid condition
[ξ, F
. η] = F . [ξ, η] ++
.
aT (ξ)(F ) η.
(28)
Theorem 5.1. Let A be a Lie algebroid on P . Then aT = J ◦ T (a) : T A −→ T 2 P
and the bracket on ΓT P T A defined by Equation (27) make T A a Lie algebroid on
T P . The bundle projection pA : T A −→ A is a morphism of Lie algebroids over
p : T P −→ P .
LIE BIALGEBROIDS AND POISSON GROUPOIDS
13
] and aT (X)
b = a(X) ˘ for X ∈ Γ(A);
Proof. Notice first that aT (T (X)) = a(X)
the latter uses the fact that the map induced on the cores by T (a) : T A −→ T 2 P
is a : A −→ T P itself. From these and Equations (26) it follows that aT ([ξ, η]) =
b Similarly the Jacobi identity
[aT (ξ), aT (η)] holds for ξ and η of the form T (X) or X.
is easily verified on sections of these forms.
To prove the nonlinearity condition (28) for ξ = T (X), F = f ◦ p, η = Yb , we have
.
[T (X), (f ◦ p) Yb ]
=
=
[T (X), (f Y ) b ]
[X, f Y ] b
.
.
.
(f ◦ p) [X, Y ] b +
+ (a(X)(f ) ◦ p)
] ◦ p)
b
= (f ◦ p) [T (X), Y ] +
+ a(X)(f
= (f ◦ p) [T (X), Yb ] +
+ aT (T (X))(f
=
. Yb
. Yb
.
◦ p) Yb .
b F = f ◦ p, η = Yb . The remaining cases are
A similar proof applies for ξ = X,
handled by using the equation
.
.
T (f Y ) = (f ◦ p) T (Y ) +
+ fe Yb
and polarizing with respect to
ffg = (f ◦ p)e
g + fe(g ◦ p).
This proves Equation (28), and the general cases of aT ([ξ, η]) = [aT (ξ), aT (η)] and
the Jacobi identity now follow.
To prove that pA : T A −→ A is a morphism of Lie algebroids, it suffices [9] to
prove that if ξ, η ∈ ΓT P T A are projectable with ξ ∼ X, η ∼ Y , then [ξ, η] ∼ [X, Y ].
But the projectable sections of T A are precisely the linear combinations of those
of the form T (X) and Yb , and so the result follows from the defining conditions
(27).
In the case of A = T P , this construction equips the bundle T (p) : T 2 P −→ T P
with a Lie algebroid structure with anchor J : T 2 P −→ T 2 P and bracket
[ξ, η] = J[Jξ, Jη].
Remark 5.2. Let G be a Lie groupoid on P . We show in Theorem 7.1 that T (AG),
with the structure of Theorem 5.1, is isomorphic to A(T G), the Lie algebroid of
the tangent groupoid T G on T P , under a map induced by the canonical involution
on T 2 G.
We now return to consideration of a general vector bundle (A, q, P ). It is a
remarkable fact that the cotangent bundle of A has a double vector bundle structure:
T ∗A
cA
r
−−−−−−−−−−−→




y
A
A∗




y
−−−−−−−−−−−→
q
P;
q∗
(29)
14
K. C. H. MACKENZIE AND PING XU
we call (29) the cotangent dual of T A. Here (T ∗ A, cA , A) is the standard cotangent
∗
bundle of A, and the notation TX
(A) will always refer to the fibre with respect to
∗
cA . In this bundle we use standard notation, and denote the zero element of TX
(A)
∗
by e
0X .
∗
The map r : T ∗ A −→ A∗ is defined as follows. Take Φ ∈ TX
(A), where X ∈ Am .
Define
r(Φ)(Y ) = Φ(τ (X, Y ))
∗
for Y ∈ Am . Thus r(Φ) ∈ A∗m . Given Φ ∈ TX
(A), Ψ ∈ TY∗ (A) with r(Φ) = r(Ψ) ∈
∗
A∗m , define Φ +
+ Ψ ∈ TX+Y
(A) by
(Φ +
+ Ψ)(ξ +
+ η) = Φ(ξ) + Ψ(η),
where ξ ∈ TX (A), η ∈ TY (A), and T (q)(ξ) = T (q)(η). Similarly, define
.
.
(λ Φ)(λ ξ) = λΦ(ξ),
for λ ∈ R and ξ ∈ TX (A). The zero element of r−1 (φ), where φ ∈ A∗m , is e
0rφ ∈
T0∗m (A) defined by
e
0rφ (T (0)(x) + Y ) = φ(Y )
for x ∈ Tm (P ), Y ∈ Am .
It is straightforward to verify that (29) is a double vector bundle, and that its core
∗
(P ), the corresponding core element ω is ω(T (0)(x) + Y ) =
is T ∗ P . Given ω ∈ Tm
ω(x), for x ∈ Tm (P ), Y ∈ Am . The core exact sequence over A is
r!
q ! T ∗ P >−−−> T ∗ A −−− q ! A∗ ,
(30)
and this is the dual of the core exact sequence (21) for T A and pA . The map
+ ω; this is the pullback of ω to A
q ! T ∗ P → T ∗ A sends (X, ω) ∈ A × T ∗ P to e
0∗X +
P
at the point X. The other core exact sequence is
c!
q∗! T ∗ P >−−−> T ∗ A −−− q∗! A,
(31)
where each bundle here is over A∗ .
Similarly there is a double vector bundle
T •A
r•
T (q)∗
−−−−−−−−−−−→




y
A∗
TP




y
−−−−−−−−−−−→
q∗
p
(32)
P,
formed by dualizing the horizontal structure in (20). We refer to this as the T (q)dual of T A; the bundle (T • (A), T (q)∗ , T P ) is defined to be the dual of the vector
bundle (T A, T (q), T P ). The fibre of T (q)∗ over x ∈ T P is denoted Tx• (A). An
element f ∈ Tx• (A) is now a linear map T (q)−1 (x) −→ R, and addition and scalar
multiplication in this bundle is the standard addition and multiplication of linear
maps. The zero element in Tx• (A) is denoted e
0•x . We denote the operations in this
vector bundle by the usual symbols.
LIE BIALGEBROIDS AND POISSON GROUPOIDS
15
The map r• : T • A −→ A∗ is defined as follows. Take f : T (q)−1 (x) −→ R, where
x ∈ Tm (P ), and define
X ∈ Am .
r• (f)(X) = f(T (0)(x) + X),
Given f ∈
by
Tx• (A),
s∈
Ty• (A)
•
with r• (f) = r• (s) = φ ∈ A∗m , define f +
+ s ∈ Tx+y
(A)
(f +
+ s)(ξ + η) = f(ξ) + s(η),
where ξ, η ∈ T (A) have T (q)(ξ) = x, T (q)(η) = y and pA (ξ) = pA (η).
Similarly, scalar multiplication of f as above by λ ∈ R is given by
.
ξ ∈ T (q)−1 (x).
(λ f)(λξ) = λf(ξ),
The zero element of r•−1 (φ), where φ ∈ A∗m , is e
0•φ : T (q)−1 (0Tm ) → R, defined by
e
0•φ (e
0X +
+ Y ) = φ(Y ),
X, Y ∈ Am .
•
The core of T A can be canonically identified with A∗ , with φ ∈ A∗m corresponding to φ ∈ T • A where
φ(e
0X +
+ Y ) = φ(X),
X, Y ∈ Am .
The core sequence for T • A over T P is now
(p!A )∗
r•! =υ ∗
p! A∗ >−−−> T • A −−− p! A∗ ,
(33)
and this is the dual of the T (q)-sequence (22) for T A. The core sequence over A∗
is
χ
T (q)!∗
q∗! A∗ >−−−> T • A −−− q∗! T P,
(34)
+ ψ)(e
0•φ +
0X +
+ Y ) = φ(Y ) + ψ(X) for φ, ψ ∈
+ ψ and (e
where χ(φ, ψ) = e
0•φ +
∗
Am , X, Y ∈ Am .
If A is the trivial bundle P × V , we can write T • A −→ T P as T P × V ∗ × V ∗ −→
T P where f = (x, φ, φ0 ) acts on ξ = (x, X, X 0 ) ∈ T P × V × V by
h(x, φ, φ0 ), (x, X, X 0 )i = hφ, Xi + hφ0 , X 0 i.
The operations in this bundle are accordingly similar to those given in equations
(24) for T P × V × V −→ T P .
The operations in r• : T P × V ∗ × V ∗ −→ P × V ∗ , (x, φ, φ0 ) 7→ (px, φ0 ), are given
by
(x, φ, φ0 ) +
+ (y, ψ, φ0 )
.
=
(x + y, φ + ψ, φ0 ),
0
t (x, φ, φ ) = (tx, tφ, φ0 ),
e
0•(m,φ) = (0Tm , 0, φ),
(35)
where m ∈ P, φ, φ0 , ψ ∈ V ∗ and x, y ∈ T P have p(x) = p(y). Given (m, φ) ∈ P ×V ∗ ,
the corresponding core element is
(m, φ) = (0Tm , φ, 0).
Both (29) and (32) are instances of the general dualization process for double
vector bundles of Pradines [21].
Proposition 5.3. For any vector bundle (A, q, P ), the double vector bundle (32)
is canonically isomorphic to the tangent double vector bundle of A∗ , by an isomorphism preserving the side bundles A∗ and T P , and the core A∗ .
16
K. C. H. MACKENZIE AND PING XU
Proof. Applying the tangent functor to the canonical pairing h , i : A∗ × A −→
P
P × R, we obtain a pairing T (A∗ ) × T A −→ T P × T R, denoted hh , ii, given
TP
explicitly by
d
hhX, ξii =
(36)
hφt , Xt i ,
dt
0
d
d
where X = dt
φt 0 ∈ T (A∗ ), ξ = dt
Xt 0 ∈ T A have T (q∗ )(X) = T (q)(ξ); we can
thus arrange that q∗ (φt ) = q(Xt ) for t near zero. It is straightforward to prove that
this is nondegenerate, and therefore gives an isomorphism I : T (A∗ ) −→ T • (A) of
the vector bundle structures over T P .
We need to prove that I is also a morphism of the vector bundle structures over
A∗ ; that is, that r• ◦ I = pA∗ , I(X + Y) = I(X) +
+ I(Y) and I(tX) = t I(X) for
suitable X, Y ∈ T (A∗ ) and t ∈ R. This is a local question, and so we can assume
that A is a trivial bundle P × V . The tangent double vector bundle structure for
A = P × V is given in (24) and (25), and the structure of T (A∗ ) is similar. Take
ξ = (x, X, X 0 ) ∈ T P × V × V and X = (x, f, f 0 ) ∈ T P × V ∗ × V ∗ . Then
d
d
0 0 (mt , f + tf ) , (mt , X + tX )
hhX, ξii =
dt
0 dt
0
d
0
0
2 0
0 (f (X) + tf (X) + f (tX ) + t f (X ))
=
dt
0
.
=
f 0 (X) + f (X 0 ).
It follows that I is given locally by I(x, f, f 0 ) = (x, f 0 , f ). Now it is straightforward,
comparing equations (25) and (35), to see that I is a vector bundle morphism over
A∗ .
It is also clear from the local description that I is an isomorphism of double
vector bundles, and preserves the side bundles and the core.
Although T • (A) thus does not provide a new structure, we will find the alternative description of T (A∗ ) which it provides useful.
Remark 5.4. Consider the canonical involution J : T 2 P −→ T 2 P . Dualizing over
T P we obtain a vector bundle morphism
•
T (T P )
T (p)∗
J∗
−−−−−−−−−−−→




y
TP
T ∗ (T P )




y
==============
cT P
T P,
which is easily seen to be a morphism of double vector bundles preserving T P
and T ∗ P . Composing with I gives an isomorphism of double vector bundles
T (T ∗ P ) −→ T ∗ (T P ), also preserving the side bundles, which we denote J 0 . For a
local description see, for example, [6].
We now show that T ∗ (A∗ ) and T ∗ (A) are isomorphic as double vector bundles.
LIE BIALGEBROIDS AND POISSON GROUPOIDS
17
Theorem 5.5. For any vector bundle (A, q, P ), the cotangent dual T ∗ (A) (29)
is canonically isomorphic to the cotangent dual T ∗ (A∗ ) of A∗ , by a double vector
bundle isomorphism which preserves the side bundles A and A∗ .
Proof. Write K = A∗ × A and define F : K −→ R to be the pairing. By a conP
struction of Tulczyjew [23], there is a Lagrangian submanifold N ⊆ T ∗ (A∗ × A) =
T ∗ (A∗ ) × T ∗ (A) defined to consist of all elements (F, Φ), where (F; ψ, Y ; m) ∈
T ∗ (A∗ ) and (Φ; X, φ; m) ∈ T ∗ (A) have (ψ, X) ∈ A∗ × A, and such that
P
h(F, Φ), (X, ξ)i = hdF, (X, ξ)i
∗
(37)
∗
for all (X, ξ) ∈ T (A × A) = T (A ) × T (A) compatible with (F, Φ). Here the comP
TP
patibility condition forces X and ξ to have the forms (X; ψ, x : m) and (ξ; X, x; m)
respectively.
Now the pairing on the left of Equation (37) is simply hF, Xi + hΦ, ξi, and
hdF, (X, ξ)i = hhX, ξii. Thus the condition (37) becomes
hF, Xi + hΦ, ξi = hhX, ξii.
(38)
We claim that N is the graph of the isomorphism we seek. To prove that it is
a graph, we must show that, given F ∈ T ∗ (A∗ ), there exists a unique Φ ∈ T ∗ (A)
satisfying (38). Since F and Φ must lie above the same point m ∈ P , we can work
locally. Assume, therefore, that A = P × V is a trivial bundle, and write
F = (χ, ψ, Y ) ∈ T ∗ P × V ∗ × V ;
Φ = (ω, X, φ) ∈ T ∗ P × V × V ∗ ;
X = (x, ψ, θ) ∈ T P × V ∗ × V ∗ ;
ξ = (x, X, Z) ∈ T P × V × V.
Then hF, Xi = hχ, xi + hθ, Y i; hΦ, ξi = hω, xi + hφ, Zi; hhX, ξii = hψ, Zi + hθ, Xi,
and (38) becomes
hχ, xi + hθ, Y i + hω, xi + hφ, Zi = hψ, Zi + hθ, Xi.
It is clear that, given χ, ψ and Y , the unique ω, X and φ for which this equation is
satisfied for all x, θ and Z are ω = −χ, X = Y and φ = ψ. This proves that N is the
graph of a map R : T ∗ (A∗ ) −→ T ∗ (A). It now follows from the local representation
R : (χ, ψ, Y ) 7→ (−χ, Y, ψ) that R is a diffeomorphism of double vector bundles, and
preserves A and A∗ .
It is clear from the local representation that the map of the cores T ∗ P −→ T ∗ P
induced by R is the negative of the identity. Since N is Lagrangian, it also follows
that R is anti-symplectic with respect to the canonical symplectic structures on the
cotangent bundles.
Lastly in this section, we make precise the relationship between the tangent Lie
algebroid of Theorem 5.1 and the tangent Poisson structures of Courant [6]. We
first need to recall the duality between Lie algebroids and (what we call) Poisson
vector bundles; see [4], [5] for further details.
Let q : A −→ P be a Lie algebroid with anchor a : A −→ T P . For any X ∈ Γ(A),
write lX for the corresponding linear function φ 7→ hφ, Xi on A∗ . Then the dual
bundle q∗ : A∗ −→ P has a Poisson structure characterized by the following three
equations:
{lX , lY } = l[X,Y ] ,
{lX , q∗∗ f } = q∗∗ (a(X)f ),
∞
{q∗∗ f, q∗∗ g} = 0,
(39)
where X, Y ∈ Γ(A) and f, g ∈ C (P ). Call this the Poisson structure dual to the
Lie algebroid structure on A.
18
K. C. H. MACKENZIE AND PING XU
Conversely, consider a vector bundle E −→ P . Denote the C ∞ (P )-module of
∞
∞
functions E −→ R which are fibrewise linear by C`in
(E). Thus C`in
(E) ∼
= Γ(E ∗ ).
If E has a Poisson structure such that the Poisson bracket of any two elements of
∞
∞
C`in
(E) lies in C`in
(E), then we call E a Poisson vector bundle. Given a Poisson
vector bundle q : E −→ P , define a bracket on Γ(E ∗ ) by l[X,Y ] = {lX , lY }, where
X, Y ∈ Γ(E ∗ ). Further, given X ∈ Γ(E ∗ ), the Hamiltonian vector field HlX =
{lX , −} ∈ ΓT E projects under q to a vector field on P which we denote by a(X).
The resulting map a : E ∗ −→ T P and the bracket [ , ] make E ∗ a Lie algebroid on
P.
Next, recall the tangent Poisson structures of [6]. Given a manifold P , the
module of functions on T P is generated by all fe and all f ◦ p for f ∈ C ∞ (P ); here
fe = lδf and p : T P −→ P is the bundle projection. If P has a Poisson bracket { , }
then the tangent Poisson structure on T P is characterized by
^
{fe, ge} = {f,
g},
{fe, g ◦ p} = {f, g} ◦ p,
{f ◦ p, g ◦ p} = 0,
(40)
∞
for f, g ∈ C (P ). This makes T P −→ P a Poisson vector bundle, and as such it
is dual to the Lie algebroid structure on T ∗ P −→ P of [4].
Theorem 5.6. (i) If E −→ P is a Poisson vector bundle, then the tangent Poisson
structure on T E makes T E −→ T P a Poisson vector bundle.
(ii) Let (A, q, P ) be a Lie algebroid, and give T A −→ T P the tangent Lie algebroid structure of Theorem 5.1, and A∗ −→ P the dual Poisson vector bundle
structure of (39). Then I : T (A∗ ) −→ T • (A) is a Poisson isomorphism with respect to the tangent Poisson structure on T (A∗ ) induced from the Poisson structure
on A∗ , and the dual Poisson structure on T • (A) induced from the tangent Lie
algebroid structure on T A −→ T P .
Proof. The two statements have essentially the same proof, and we formulate it
for the second statement. First observe that if X ∈ Γ(A), then lf
X and lX ◦ p∗ ,
where p∗ : T (A∗ ) −→ A∗ is the bundle projection, are fibrewise linear functions
T (A∗ ) −→ R with respect to the bundle T (A∗ ) −→ T P ; further, the lf
X and lX ◦ p∗
∞
(T (A∗ )). Now it is easy to verify that, for all X ∈ Γ(A),
generate C`in
lf
X = lT (X) ◦ I,
lX ◦ p∗ = lX
b ◦ I,
where l on the right-hand sides refers to the duality between T A −→ T P and
T • A −→ T P .
Given X, Y ∈ Γ(A), and using (39), (40) and (27),
^
e
^
{lT (X) ◦ I, lT (Y ) ◦ I} = {lf
X , lY } = {lX , lY } = l[X,Y ]
= lT ([X,Y ]) ◦ I = l[T (X),T (Y )] ◦ I = {lT (X) , lT (Y ) } ◦ I,
and similarly for the other two conditions.
It is interesting to note that, for a Lie algebroid (A, q, P ), the double vector bundle (29) has Poisson structures on both vertical bundles, and Lie algebroid structures on both horizontal bundles, whilst (32) has Poisson structures horizontally
and Lie algebroid structures vertically. The relationship between these structures
will be dealt with fully elsewhere. Here we just note the following, whose proof is
straightforward.
LIE BIALGEBROIDS AND POISSON GROUPOIDS
19
Proposition 5.7. Let (A, q, P ) be a Lie algebroid. Then in the double vector bundle
T ∗ (A∗ )
−−−−−−−−−−−→




y




y
A∗
A
−−−−−−−−−−−→
(41)
P
the structure maps for the horizontal vector bundles are Lie algebroid morphisms
with respect to the given structure on A and the cotangent structure on T ∗ (A∗ ) in#
∗
∗
duced by the Poisson structure on A∗ . In particular, the anchor πA
∗ : T (A ) −→
∗
T (A ) is a double vector bundle morphism with respect to a : A −→ T P and
id : A∗ −→ A∗ , and induces −a∗ : T ∗ P −→ A∗ on the cores.
We will see that this structure plays a crucial role when (A, A∗ ) is a Lie bialgebroid.
6. An equivalent definition
The purpose of this section is to give an equivalent definition, 6.2 below, of
Lie bialgebroids in terms of Lie algebroid morphisms. The importance of this will
become apparent in §8. In fact, it is this equivalent definition that mainly motivates
our present work.
The notion of Lie algebroid morphism, and its properties as an infinitesimal
analogue of Lie groupoid morphism, were studied at length in [9]. In applications,
the following equivalent definition is useful, particularly when dealing with Poisson
geometry.
Proposition 6.1. Let A and B be Lie algebroids on bases P and Q and consider
a vector bundle morphism F : A −→ B, f : P −→ Q. Let C be the submanifold of
∗
∗
A∗ × B consisting of all elements (φ, ψ) ∈ A∗ × B such that hφ, Xi = hψ, F (X)i
for all X ∈ A compatible with φ. Then F is a Lie algebroid morphism if and only
∗
if C is a coisotropic submanifold of A∗ × B , where A∗ and B ∗ are equipped with
the Poisson structures dual to the Lie algebroids A and B, respectively.
Proof. Evidently C is the annihilator in A × B of the graph of (F, f ). Now the
result follows from the facts that (i), the vector bundle morphism (F, f ) is a Lie
algebroid morphism if and only if its graph is a Lie subalgebroid of A × B [9], and
(ii), if C −→ P is a Lie algebroid, and D −→ Q is a vector subbundle, then D is a
Lie subalgebroid of C if and only if the annihilator D⊥ of D in C ∗ is coisotropic in
C ∗.
The main theorem of this section is the following.
Theorem 6.2. Suppose that q : A −→ P is a Lie algebroid such that its dual vector
bundle q∗ : A∗ −→ P also has a Lie algebroid structure. Let a, a∗ be their anchors.
20
K. C. H. MACKENZIE AND PING XU
Then (A, A∗ ) is a Lie bialgebroid if and only if
T ∗ (A∗ )
Π
−−−−−−−−−−−→




y




y
A∗
TA
−−−−−−−−−−−→
a∗
(42)
TP
is a Lie algebroid morphism, where the domain T ∗ (A∗ ) −→ A∗ is the cotangent
Lie algebroid induced by the Poisson structure on A∗ , the target T A −→ T P is
the tangent Lie algebroid of A, and Π : T ∗ (A∗ ) −→ T A is the composition of the
#
isomorphism R : T ∗ A∗ −→ T ∗ A of Theorem 5.5 with πA
: T ∗ A −→ T A.
The proof of Theorem 6.2 will take us to the end of the section. We assume until
then that A and A∗ satisfy the hypotheses of 6.2. The three preliminary results
which follow are needed in the course of the proof.
Lemma 6.3. Given (ξ; Xm , x; m) ∈ T A and (X; φm , x; m) ∈ T (A∗ ), let X ∈ Γ(A)
and φ ∈ Γ(A∗ ) be any sections taking the values Xm and φm at m. Then
hhX, ξii = X(lX ) + ξ(lφ ) − x(hφ, Xi).
∗
(P ) and X ∈ Am , denote the
The proof is straightforward. Given ω ∈ Tm
pullback of ω to A at the point X by q ∗ (X, ω).
Proposition 6.4. For X ∈ Γ(A) and φ ∈ Γ(A∗ ),
R(δlX (φm )) = δlφ (Xm ) − q ∗ (Xm , δhφ, Xi),
where δlX (φm ) ∈ Tφ∗m (A∗ ) is the value of the 1-form δlX on A∗ at φm , and similarly
for δlφ (Xm ).
Proof. For any m ∈ P , obviously (φm , Xm ) ∈ A∗ × A = K, in the notation of the
P
proof of Theorem 5.5. It suffices to show that
(δlX (φm ), δlφ (Xm ) − q ∗ (Xm , δhφ, Xi)) ∈ N.
Let (X, ξ) be any tangent vector of K at the point (φm , Xm ). Then
hX, δlX (φm )i + hξ, δlφ (Xm ) − q ∗ (Xm , δhφ, Xi)i =
X(lX )(φm ) + ξ(lφ )(Xm ) − (T (q)(ξ))(hφ, Xi),
and this is equal to hhX, ξii by Lemma 6.3. The conclusion now follows from the
proof of Theorem 5.5.
Corollary 6.5. Let X ∈ Γ(A) be any section. Then for any (φm , ψm ) ∈ A∗ × A∗ ,
P
with q∗ (φm ) = q ∗ (ψm ) = m,
πA (R(δlX (φm )), R(δlX (ψm ))) = −(d∗ X)(φm ∧ ψm ),
where πA denotes the Poisson tensor on A induced from the Lie algebroid A∗ .
LIE BIALGEBROIDS AND POISSON GROUPOIDS
21
Proof. Let φ ∈ Γ(A∗ ) be any section through the point φm and ψ ∈ Γ(A∗ ) any
section through the point ψm . Then,
πA (R(δlX (φm )), R(δlX (ψm )))
= πA (δlφ (Xm ) − q ∗ (Xm , δhφ, Xi), δlψ (Xm ) − q ∗ (Xm , δhψ, Xi))
= h[φ, ψ], Xi − Lφ (hψ, Xi) + Lψ (hφ, Xi)
= −(d∗ X)(φm ∧ ψm ),
which implies the desired result.
Throughout this section, let C denote the annihilator of the graph of (Π, a∗ ).
Thus C is the subset of T A∗ × T A∗ consisting of all elements (X, Y) ∈ Tφ A∗ × Tψ A∗
such that a∗ (φ) = T (q∗ )(Y) and hX, Fi = hhY, Π(F)ii for all F ∈ Tφ∗ A∗ . We first
give another description of C, which is more useful in practice. For this purpose,
we need to introduce the following function FX .
For any X ∈ Γ(A), let FX be the function on T A∗ × T A∗ defined by
FX (X, Y) = X(lX ) − Y(lX ) + fd∗ X (φ, ψ),
where fd∗ X is any function on A∗ ×A∗ extending the function (φ, ψ) 7→ (d∗ X)(φ ∧ ψ)
on A∗ × A∗ .
P
Proposition 6.6. Let (X, Y) ∈ Tφm A∗ × Tψm A∗ . Then (X, Y) ∈ C if and only if
the three conditions
a∗ (φm ) = T (q∗ )(Y),
a∗ (ψm ) = T (q∗ )(X),
FX (X, Y) = 0,
X ∈ Γ(A),
hold.
We first need a lemma.
Lemma 6.7. Suppose that (φm , ψm ) ∈ A∗ × A∗ with m = q∗ (φm ) = q∗ (ψm ). Take
P
∗
ω ∈ Tm
(P ). Then for the pullbacks q∗∗ (φm , ω) and q∗∗ (ψm , ω) of ω to φm , ψm ∈ A∗ ,
we have
πA (R(q∗∗ (φm , ω)), R(q∗∗ (ψm , ω))) = ha∗ (ψm ) − a∗ (φm ), ωi.
0∗φm +
+
Proof. Recall from the exact sequence (30), applied to A∗ , that q∗∗ (φm , ω) = e
∗
∗
ω ∈ Tφm (A ). Since R is a double vector bundle morphism reversing the core, it
follows that R(e
0∗φm +
+ ω) = e
0rφm − ω. Now e
0rφm = δlφ (0m ), where φ ∈ Γ(A∗ ) is any
section passing through φm . So
πA (R(q∗∗ (φm , ω)), R(q∗∗ (ψm , ω)))
=
πA (δlφ (0m ) − ω, δlψ (0m ) − ω)
=
l[φ,ψ] (0m ) − ha∗ (φ), ωi + ha∗ (ψ), ωi,
whence the result. Note that ω is the pullback of ω to 0∗m ∈ A∗m .
Proof of Proposition 6.6. It is clear that Tφ∗m A∗ is spanned by the covectors of the
∗
form (δlX )(φm ), for X ∈ Γ(A), and those of the form q∗∗ (φm , ω), for ω ∈ Tm
P . So
(X, Y) ∈ C if and only if the three conditions
a∗ (φm ) = T (q∗ )(Y),
hX, (δlX )(φm )i = hhY, Π(δlX (φm ))ii,
hX, q∗∗ (φm , ω)i = hhY, Π(q∗∗ (φm , ω))ii,
∗
hold for all X ∈ Γ(A), ω ∈ Tm
(P ).
22
K. C. H. MACKENZIE AND PING XU
Using Lemma 6.3 and letting ψ ∈ Γ(A∗ ) be any section passing through ψm , we
have
hhY, Π(δlX (φm ))ii = Y(lX ) + Π(δlX (φm ))(lψ ) − T (q∗ )(Y)(hψ, Xi).
Now, using Proposition 6.4 and Corollary 6.5,
Π(δlX (φm ))(lψ )
=
πA (R(δlX (φm )), R(δlX (ψm )))
+ πA (R(δlX (φm )), q ∗ (Xm , δhψ, Xi))
=
−d∗ X(φm ∧ ψm ) + T (q∗ )(Y)hψ, Xi,
since a∗ (φm ) = T (q∗ )(Y). Thus the second condition is equivalent to FX (X, Y) = 0
for all X ∈ Γ(A).
Similarly, using Lemma 6.7,
hhY, Π(q∗∗ (φm , ω))ii
= Y(l0 ) + Π((q∗∗ (φm , ω))(lψ ) − T (q∗ )(Y)(hψ, 0i)
= πA (R(q∗∗ (φm , ω)), δlψ (0m ) + ω) − πA (R(q∗∗ (φm , ω)), ω)
= πA (R(q∗∗ (φm , ω)), R(q∗∗ (ψm , ω))
− πA (R(q∗∗ (φm , ω)), R(q∗∗ (0m , ω)))
= ha∗ (ψm ) − a∗ (φm ), ωi − ha∗ (0m ) − a∗ (φm ), ωi
= ha∗ (ψm ), ωi,
and so the third condition is equivalent to a∗ (ψm ) = T (q∗ )(X).
The next result is a corollary of Proposition 6.6.
Corollary 6.8. Let π be the projection of T ∗ A∗ × T ∗ A∗ onto A∗ × A∗ . Then
π(C) = A∗ × A∗ .
P
For any ω ∈ Ω1 (P ), define functions Gω and Hω on T A∗ × T A∗ by:
Gω (X, Y) = ha∗ (φ) − T (q∗ )Y, ωi,
Tφ∗ (A∗ ),
and
Hω (X, Y) = ha∗ (ψ) − T (q∗ )X, ωi,
Tψ∗ (A∗ ).
where X ∈
Y∈
Proposition 6.6 shows that C is the set of common
zeros of the three families of functions FX , Gω , Hω , for X ∈ Γ(A), ω ∈ Ω1 (P ).
The next step is to calculate the Poisson brackets of these functions on T A∗ × T A∗ .
Throughout the rest of the section, A∗ is understood to have the Poisson structure
dual to the Lie algebroid structure on A, the opposite Poisson structure is denoted
A∗ , and T A∗ and T A∗ have the corresponding tangent Poisson structures.
Theorem 6.9. Take X, Y ∈ Γ(A). Then for any (X, Y) ∈ Tφ A∗ × Tψ A∗ ,
{FX , FY }(X, Y) − F[X,Y ] (X, Y) = (LX d∗ Y − LY d∗ X − d∗ [X, Y ])(φ ∧ ψ),
where the Poisson bracket on the left hand side is with respect to the product Poisson
structure on T A∗ × T A∗ .
Again we start with a lemma. Let D ∈ Γ(∧2 A) be any bi-section of A. Thus D
defines a function on A∗ × A∗ by (φ, ψ) 7→ D(φ ∧ ψ). Let fD be any extension of
D to A∗ × A∗ .
P
1
2
Lemma 6.10. For any X ∈ Γ(A), let lX
and lX
denote the linear functions on
∗
1
2
∗
A × A defined by lX (φ, ψ) = hφ, Xi, and lX (φ, ψ) = hψ, Xi, for any (φ, ψ) ∈
A∗ × A∗ respectively. Then
1
2
{lX
, fD }(φ, ψ) − {lX
, fD }(φ, ψ) = (LX D)(φ ∧ ψ),
(43)
LIE BIALGEBROIDS AND POISSON GROUPOIDS
23
where the Poisson bracket on the left hand side is with respect to the product Poisson
structure on A∗ × A∗ .
Proof. Without loss of generality, we assume that D = Y1 ∧ Y2 for Y1 , Y2 ∈ Γ(A)
and fD (φ, ψ) = Y1 (φ)Y2 (ψ) − Y1 (ψ)Y2 (φ) for φ, ψ ∈ A∗ . Let φ and ψ also denote
any sections of A∗ through the points φ and ψ respectively. Then,
(LX D)(φ ∧ ψ)
=
−D(LX (φ ∧ ψ)) + LX (D(φ ∧ ψ))
=
−(ιφ D) · (LX ψ) + (ιψ D) · (LX φ) + LX (D(φ ∧ ψ))
=
[X, ιφ (D)](ψ) − [X, ιψ (D)](φ) − LX (D(φ ∧ ψ))
=
h[X, (Y1 · φ)Y2 − (Y2 · φ)Y1 ], ψi
− h[X, (Y1 · ψ)Y2 − (Y2 · ψ)Y1 ], φi
− LX (D(φ ∧ ψ))
=
(Y1 · φ)([X, Y2 ] · ψ) − (Y2 · φ)([X, Y1 ] · ψ)
− (Y1 · ψ)([X, Y2 ] · φ) + (Y2 · ψ)([X, Y1 ] · φ).
On the other hand, it can be easily checked, by definition, that the left hand side
of Equation (43) is exactly the same as above.
Proof of Theorem 6.9. It follows from a straightfoward computation, using Equations (39) and (40), that
{FX , FY }(X, Y)
=
1
X(l[X,Y ] ) − Y(l[X,Y ] ) + {lX
, fd∗ Y }(φ, ψ)
2
−{lX
, fd∗ Y }(φ, ψ) − {lY1 , fd∗ X }(φ, ψ) + {lY2 , fd∗ X }(φ, ψ),
where the brackets on the right hand side are with respect to the product Poisson
structure on A∗ × A∗ . The result now follows immediately from Lemma 6.10. Theorem 6.11. For any ω, θ ∈ Ω1 (P ),
{Gω , Gθ } = 0,
{Hω , Hθ } = 0,
and for X ∈ Tφ (A∗ ), Y ∈ Tψ (A∗ ),
{Gω , Hθ }(X, Y) = −πP (ω, θ)(q∗ (φ)) + πP (ω, θ)(q∗ (ψ)).
Proof. Clearly,
{Gω , Gθ }(X, Y)
=
{ha∗ (φ), ωi − hT (q∗ )Y, ωi, ha∗ (φ), θi − hT (q∗ )Y, θi}
=
{hT (q∗ )Y, ωi, hT (q∗ )Y, θi}
= {lq∗∗ ω , lq∗∗ θ }
= l{q∗∗ ω,q∗∗ θ}
=
0.
The bracket in the second-last line is the Lie algebroid bracket in the cotangent
Lie algebroid T ∗ A∗ . Note we are using the fact that a tangent Poisson structure is
dual to the corresponding cotangent Lie algebroid.
24
K. C. H. MACKENZIE AND PING XU
The proof that {Hω , Hθ } = 0 is similar. For the third identity, we have
{Gω , Hθ }(X, Y)
= −{ha∗ (φ), ωi, hT (q∗ )X, θi} − {hT (q∗ )Y, ωi, ha∗ (ψ), θi}
#
#
∗
∗
= πA
∗ (q∗ θ)(ha∗ (φ), ωi) + πA∗ (q∗ ω)(ha∗ (ψ), θi)
= πA∗ (q∗∗ θ, δla∗∗ ω )(φ) + πA∗ (q∗∗ ω, δla∗∗ θ )(ψ)
= −ha ◦ a∗∗ (ω), θi(q∗ (φ)) − ha ◦ a∗∗ (θ), ωi(q∗ (ψ))
= −πP (ω, θ)(q∗ (φ)) + πP (ω, θ)(q∗ (ψ)).
Note that in the second equality, we have used the fact that the Poisson structure
on the second factor is the tangent Poisson structure of A∗ .
Theorem 6.12. Suppose that (A, A∗ ) is a Lie bialgebroid. For any X ∈ Γ(A) and
ω ∈ Ω1 (P ),
{FX , Gω } = Gτ
and
{FX , Hω } = Hτ ,
where τ = δha(X), ωi + ιa(X) δω.
Proof. It is clear that
{FX , Gω }(X, Y)
= {X(lX ), ha∗ (φ), ωi} + {Y(lX ), hT (q∗ )Y, ωi}
− {fd∗ X (φ, ψ), hT (q∗ )Y, ωi}
=
#
[πA
∗ (δlX )](ha∗ (φ), ωi)(φ) + {lδlX , lq ∗ ω }(Y)
∗
#
∗
− (πA
∗ (q∗ ω))(fd∗ X (φ, ψ))
According to Corollary 3.9, the first term is easily seen to be
[X, a∗∗ ω](φ)
=
(La∗ ω X)(φ) + (a∗∗ (ιa(X) δω))(φ)
= d∗ (ιa∗ ω X)(φ) + (ιa∗ ω d∗ X)(φ) + (a∗∗ (ιa(X) δω))(φ)
= ha∗ (φ), τ i + (d∗ X)((a∗ ω) ∧ φ).
Using the standard formula for the bracket of 1-forms on a Poisson manifold (compare (19)), the second term becomes (note that the minus signs arise from the
opposite Poisson structure on the second factor)
Y
{δlX , q∗∗ ω}
=
#
−Y(πA∗ (δlX , q∗∗ ω)) − [(πA
∗ δlX )
=
#
−hT (q∗ )Y, δha(X), ωii − (δω)(T (q∗ )(πA
∗ δlX ), T (q∗ )Y)
=
−hT (q∗ )Y, δha(X), ωii − (δω)(a(X), T (q∗ )Y)
=
−hT (q∗ )Y, δha(X), ωii − hT (q∗ )Y, ιa(X) δωi
=
−hT (q∗ )Y, τ i,
q∗∗ δω](Y)
#
where the third equality uses the identity T (q∗ )(πA
∗ δlX ) = a(X), which follows
from Proposition 5.7.
#
∗
∗
Finally, πA
∗ (q∗ ω) is the core element −a (ω), again by 5.7, and the third term
∗
is therefore (d∗ X)(φ ∧ (a ω)). The first equation now follows immediately. The
second can be checked similarly.
Proof of Theorem 6.2. Suppose that (Π, a∗ ) is a Lie algebroid morphism. Then C
is a coisotropic submanifold in T A∗ × T A∗ , by Proposition 6.1, and so Theorem 6.9
implies that (d∗ [X, Y ] − LX d∗ Y + LY d∗ X)(φ ∧ ψ) = 0, for all (Xφ , Yψ ) ∈ C. From
Corollary 6.8 it therefore follows that
d∗ [X, Y ] = LX d∗ Y − LY d∗ X
LIE BIALGEBROIDS AND POISSON GROUPOIDS
25
identically on A∗ × A∗ , for all X, Y ∈ Γ(A). For the other direction, we note from
P
Proposition 6.6 that the space of functions vanishing on C is generated by those of
the forms FX , Gω , Hθ for X ∈ Γ(A) and ω, θ ∈ Ω1 (P ). Thus, by Theorems 6.9,
6.11 and 6.12, it is a Poisson subalgebra, and so C is a coisotropic submanifold. Example 6.13. If A is a Lie algebra g such that its dual g∗ is also a Lie algebra,
the tangent Poisson structure T g∗ is known to be Poisson diffeomorphic to the Lie
Poisson strcture of g ×
7 g, the semi-direct product of the Lie algebra with itself [6].
It is easily seen that in this case the submanifold C = {(τ, [τ, θ] + ω, θ, ω)|τ, θ, ω ∈
g∗ } ⊂ T g∗ × T ḡ∗ , where T g∗ is identified with g∗ × g∗ . One can check directly in
fact that C is coisotropic if and only if (g, g∗ ) is a Lie bialgebra.
7. Tangent groupoids and cotangent groupoids
In this section we present several canonical isomorphisms and some other basic
results for the Lie algebroids of tangent and cotangent groupoids. These are needed
in the final section and we expect them to be of value in other work also. In the
construction of the Lie algebroids of Lie groupoids, we follow the conventions of
[17] as modified in [18]; a different approach is given in [4].
Throughout the section we consider a fixed Lie groupoid G on base P . Then T G
is a Lie groupoid on base T P with source and target T (α), T (β) : T G −→ T P , and
composition • defined by ξ • η = T (κ)(ξ, η) where κ is the composition in G. With
this structure (T G; G, T P ; P ) is a VB-groupoid [21], [18]
TG
pG
−
−
−−
−−
−−
−−
−−
−−
−−
−−
−
−−→
−→




y
TP




y
−
−
−−
−−
−−
−−
−−
−−
−−
−−
−
−−→
−→
G
p
(44)
P;
that is, each of the groupoid structure maps is a vector bundle morphism and the
double source map (pG , T (α)) : T G −→ G × T P is a surjective submersion.
P
Taking the Lie algebroid of the horizontal structure in (44) gives a double vector
bundle
qT G
AT G −−−−−−−−−−−→ T P
A(pG )




y
AG




y
−−−−−−−−−−−→
q
p
(45)
P.
The structure in AT G −→ T P is the standard structure in the Lie algebroid of
a Lie groupoid, and is denoted with the usual symbols; the zero in the fibre over
x ∈ T P is denoted e
0x . The structure in AT G −→ AG is obtained by applying the
for the
Lie functor A to the operations in T G −→ G and we write +
+ , and
operations in AT G −→ AG. Thus the projection A(pG ) is the result of applying
the Lie functor to the morphism of groupoids pG : T G −→ G and the addition
.
26
K. C. H. MACKENZIE AND PING XU
+
+ is similarly defined by A +
+ B = A(+)(A, B). The zero above X ∈ AG is
A(0)(X). It is straightforward to verify that this is a vector bundle, and that its
core is AG −→ P .
The first result is the prototype of many similar results, and so we prove it in
detail.
Theorem 7.1. Let G be a Lie groupoid on base P . Then there is a canonical
isomorphism of double vector bundles jG : T AG −→ AT G, where AT G is as above
and T AG is the tangent double vector bundle (20) of AG −→ P , which induces the
identities on the side bundles AG and T P and on the cores AG. Further, jG is
an isomorphism of Lie algebroids over T P , where T AG −→ T P has the tangent
Lie algebroid structure of Theorem 5.1 and AT G −→ T P is the Lie algebroid of
−→ T P .
T G −→
Proof. The Lie algebroid AT G is defined by a pullback diagram
AT G
ιT G
−−−−−−−−−−−→




y
T T (α) T G




y
−−−−−−−−−−−→
T (1)
TP
T G,
where ιT G is the inclusion (compare [17, III§3]), and this fits into a morphism of
double vector bundles as in Figure 1. Here we are denoting the restrictions of maps
by the same symbols. On the other hand, we can apply the tangent functor to the
pullback diagram
AG
ιG
−−−−−−−−−−−→




y
P
T αG




y
−−−−−−−−−−−→
1
G,
and obtain a morphism of double vector bundles as in Figure 2. From Lemma 1.5
in [18] we know that the two front faces of Figure 1 and Figure 2 are isomorphic
under a restriction of the canonical involution J : T 2 G −→ T 2 G. Since in both
Figure 1 and Figure 2 the top and bottom faces are pullbacks, it follows that J
restricts to the required isomorphism jG : T AG −→ AT G. That jG preserves the
side bundles and the core is now evident.
That aT G ◦ jG = JP ◦ T (aG ) follows easily by considering both aT G and T (aG )
as restrictions of T 2 (β) and using JP ◦ T 2 (β) = T 2 (β) ◦ JG . To show that jG maps
the bracket (27) on T AG −→ T P to the bracket on AT G −→ T P , we need only
−
→
consider sections of T AG of the form T (X) and Yb for X, Y ∈ ΓAG. If X denotes
the right-invariant vector field on G corresponding to X, and similarly for sections
LIE BIALGEBROIDS AND POISSON GROUPOIDS
pT G
−−−−−−−−→
AT G
TP




y

 Z

Z

Z
y
A(pG )
27
Z
Z
Z
Z
Z
Z
Z
Z
Z
AG
−−−−−−−−→
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
ι
T (1)
TG
Z
Z
Z
Z Z
Z
Z
Z
Z Z
Z
Z
Z
Z Z
Z
Z
Z
Z Z
Z
Z
Z
Z Z
pT G
Z
Z
~
Z
~ Z
Z
T (α) Z
Z
Z
T
T
G
−
−
−
−
−
−
−
−→
TG
Z
Z
ιG Z Z
Z
Z


Z
Z
Z


Z
Z
Z


T (pGZ
)


Z
Z
y
Z y
Z
Z
~
Z
T αG
−−−−−−−−→
G
Figure 1
T AG
pAG




y
AG
T (qG )
−−−−−−−−→
Z
Z
Z
TP




y
Z
Z
Z
Z
Z
Z
Z
Z
−−−−−−−−→ZZP
Z
Z
Z
Z
Z
Z
T (ιG ) Z Z
T (1)
Z
Z
Z
Z Z
Z
Z
Z
Z Z
Z
Z
Z
Z Z
Z
Z Z
Z
Z
Z
Z Z
Z
Z
T
(p
G)
Z
~
Z
~ Z
Z
Z
α
Z −−−−−−−−→ T G
Z
T (T G)Z
Z
Z
Z
ιG
Z
Z
Z


Z
Z
Z


Z
Z
Z


pT Z


G
Z
Z
y
Z y
Z
Z
~
Z
T αG
Figure 2
−−−−−−−−→
G
28
K. C. H. MACKENZIE AND PING XU
of AT G and right-invariant vector fields on T G, then one may see that
−−−−→
−−−−−−→ −
→
−
→
f
jG ◦ T (X) = X ,
jG ◦ Yb = ( Y )˘,
for all X, Y ∈ ΓAG. Now the result follows easily, using Equations (27) and (26).
Next we recall from [4], [21] the cotangent groupoid structure on T ∗ G with base
the Lie algebroid dual A∗ G. In the conventions we use here, the source α
e(ω) ∈ A∗αg G
∗
∗
e
and target β(ω)
∈ Aβg G of ω ∈ Tg G are given by
α
e(ω)(X) = ω(T (Lg )(X − T (1)(a(X))),
e
β(ω)(Y
) = ω(T (Rg )(Y )),
where X ∈ Aαg G and Y ∈ Aβg G. Here Lg and Rg are the left and right translations
∗
e
in G. If θ ∈ Th∗ G and α
e(θ) = β(ω)
then αh = βg and we define θ • ω ∈ Thg
G by
(θ • ω)(Y • X) = θ(Y ) + ω(X),
Y ∈ Th G, X ∈ Tg G.
A∗m G,
1φ (T (1)(x)+
If φ ∈
then the identity element over φ is e
1φ ∈ T1∗m G defined by e
X) = φ(X) for X ∈ Am G, x ∈ Tm (P ). It may be verified that T ∗ G is a Lie groupoid
on A∗ G, and is a symplectic groupoid with respect to the canonical symplectic
structure on T ∗ G. It is, further, a VB-groupoid with respect to the usual bundle
∗
(P )
structures on T ∗ G −→ G and A∗ G −→ P . The core is T ∗ P , where ω ∈ Tm
∗
corresponds to the core element ω ∈ T1m G given by ω(T (1)(x) + X) = ω(x + a(X))
for x ∈ Tm (P ), X ∈ Am G. Accordingly the Lie algebroid AT ∗ G has a double
vector bundle structure
qT ∗ G
AT ∗ G −−−−−−−−−−−→ A∗ G
A(cG )




y
AG




y
−−−−−−−−−−−→
q
q∗
P,
where the bundle structure in AT ∗ G −→ AG arises from application of the Lie
functor to T ∗ G −→ G. In particular the anchor e
a∗ : AT ∗ G −→ T A∗ G is a morphism
of double vector bundles over a : AG −→ T P and id : A∗ G −→ A∗ G. Note that the
core morphism of e
a∗ is a∗ : T ∗ P −→ A∗ G.
Returning to (45), we can dualize the vertical structure (as in [21]) to obtain a
double vector bundle
A• T G
A(pG )∗




y
AG
with core T ∗ P .
−−−−−−−−−−−→
A∗ G




y
−−−−−−−−−−−→
q
P,
q∗
LIE BIALGEBROIDS AND POISSON GROUPOIDS
29
Proposition 7.2. There is a canonical isomorphism iG : AT ∗ G −→ A• T G of double vector bundles which preserves the side bundles and the cores.
Proof. This is very similar to the proof of Proposition 5.3. We apply the functor A
to the pairing T ∗ G × T G −→ G × R to obtain a pairing of AT ∗ G and AT G over
G
AG. Since this is a suitable restriction of the pairing of T T ∗ G and T 2 G over T G,
it is nondegenerate. The remainder of the proof follows as before.
Now we imitate the construction in Remark 5.4. Dualizing jG : T AG −→ AT G
over AG and composing with iG , we obtain an isomorphism of double vector bundles
0
jG
: AT ∗ G −→ T ∗ AG preserving the side bundles and the cores.
Theorem 7.3. Let G be a Lie groupoid over P . Then the isomorphism of Lie
algebroids s : T ∗ A∗ G −→ AT ∗ G induced as in [4] by the symplectic groupoid struc0 −1
−→ A∗ G is equal to (jG
ture on T ∗ G −→
) ◦ R where R : T ∗ A∗ G −→ T ∗ AG is the
isomorphism of Theorem 5.5.
The isomorphism s is induced by the Poisson bundle map π # : T ∗ T ∗ G −→ T T ∗ G
corresponding to the symplectic structure δΘ on T ∗ G, where Θ is the canonical
1-form, according to ιT ∗ G ◦ s = π # ◦ βe∗ , where βe∗ : T ∗ A∗ G −→ T ∗ T ∗ G takes
1φ . Thus the main
θ ∈ Tφ∗ A∗ G to its pull-back under βe at the identity element e
work is the following result.
Lemma 7.4. Let M be any manifold, and let T ∗ M have the symplectic structure
δΘ, where Θ is the canonical 1-form on T ∗ M . Then the corresponding Poisson
bundle map π # : T ∗ T ∗ M −→ T T ∗ M is equal to (J 0 )−1 ◦ R0 , where R0 : T ∗ T ∗ M −→
T ∗ T M is the isomorphism of Theorem 5.5.
Proof. From Proposition 5.7 it follows that π # is a double vector bundle morphism
preserving T M and T ∗ M . Take F ∈ T ∗ T ∗ M with projections ω and X to T ∗ M
and T M respectively. By the definition of R0 , it suffices to prove that
hF, Xi + hJ 0 π # F, ξi = hhX, ξii
(46)
∗
for all X ∈ Tω T M, ξ ∈ TX T M with T (c)(X) = T (p)(ξ) = x ∈ T M . It is easily
seen that
hJ 0 (π # (F)), ξi = hhπ # (F), J(ξ)ii.
Using Lemma 6.3, and letting ω and X also denote sections taking the given values,
we have
hhπ # (F), J(ξ)ii = (π # F)(lx )(ω) + (J(ξ))(lω )(x) − Xhω, xi.
Suppose first that F = δlX . Since the δΘ symplectic structure on T ∗ M is dual to
the canonical Lie algebroid structure on T M , we have (π # F)(lx )(ω) = l[X,x] (ω).
Evaluating hhX, ξii in the same way, and noting hF, Xi = X(lX )(ω), it remains to
prove that
l[X,x] (ω) + J(ξ)(lω )(x) − Xhω, xi = ξ(lω )(X) − xhω, Xi.
From [1, p.122] we have
J(ξ)(lω )(x) = ξ(lω )(X) + (δω)(X, x),
and the result follows. The case where F = c∗ (ω, θ) is a pullback to ω of a 1-form
θ on M is similarly verified.
30
K. C. H. MACKENZIE AND PING XU
Proof of Theorem 7.3. Take F ∈ Tφ∗ A∗ G with projection X ∈ AG. As in the
lemma, we must prove that
hF, Xi + hj 0 (s(F)), ξi = hhX, ξii
for all X ∈ Tφ A∗ G, ξ ∈ TX AG with T (q∗ )(X) = T (q)(ξ) = x ∈ T P . As
before, we have hj 0 (s(F)), ξi = hhs(F), j(ξ)ii. We now regard s(F) as in T T ∗ G
with pT ∗ G (s(F)) = e
1φ and T (c)(s(F)) = ιG (X) ∈ T G, where ιG is the inclusion
AG −→ T G. Similarly we regard j(ξ) as in T T G with pT G (j(ξ)) = T (1)(x) and
T (p)(j(ξ)) = ιG (X). In these terms s(F) = π # (βe∗ (F)) and j(ξ) = J(T (ιG )(ξ)). So,
applying Equation (46), we have
hhs(F), j(ξ)ii
= hJ 0 (π # (βe∗ (F))), T (ιG )(ξ)i
= hhT (e
1)(X), T (ιG )(ξ)ii − hβe∗ (F), T (e
1)(X)i
and it is easily verified that
hhT (e
1)(X), T (ιG )(ξ)ii = hhX, ξii
and
hβe∗ (F), T (e
1)(X)i = hF, Xi.
This completes the proof.
8. Poisson groupoids
−→ P together with a Poisson strucA Poisson groupoid [24] is a Lie groupoid G −→
ture πG on G such that the graph Λ = {(h, g, hg) | αh = βg} of the groupoid
multiplication is a coisotropic submanifold of G × G × G. It was shown in [24]
that the manifold of identity elements of a Poisson groupoid G is coisotropic in G,
and its conormal bundle ν ∗ (P ) thereby acquires a Lie algebroid structure. This
conormal bundle may be identified with A∗ G, the dual vector bundle of AG, in a
standard way, and we will always take A∗ G with this Lie algebroid structure.
It was shown in [18, §4] that if G is a Poisson Lie group, then the cotangent bundle
T ∗ G is an LA-groupoid, and in particular the Poisson bundle map T ∗ G −→ T G is a
−→ g∗ to the group T G; this condition is equivalent
groupoid morphism from T ∗ G −→
to the twisted multiplicativity equations for the dressing transformations. The
following more general result was proved by Albert and Dazord [2]. The proof we
give here seems conceptually simpler.
−→ P be a Lie groupoid with Poisson structure π, and let
Proposition 8.1. Let G −→
AG −→ P be its Lie algebroid. Then G is a Poisson groupoid if and only if the
map
T ∗G
π#
−−−−−−−−−−−→




yy
A∗ G
TG




yy
−−−−−−−−−−−→
a∗
TP
induced by the Poisson tensor π is a groupoid morphism.
The following lemma is quite obvious.
(47)
LIE BIALGEBROIDS AND POISSON GROUPOIDS
31
Lemma 8.2. Let G1 and G2 be groupoids, and Λ1 , Λ2 their graphs of multiplication. A map φ : G1 −→ G2 is a groupoid morphism if and only if φ0 (Λ1 ) ⊆ Λ2 ,
where φ0 = φ × φ × φ : G1 × G1 × G1 −→ G2 × G2 × G2 .
Proof of Proposition 8.1. By Λ ⊂ G × G × G we denote the graph of multiplication
of the groupoid G. It is known that the graph of multiplication of the groupoid
−→ A∗ G is ν̄ ∗ Λ, which is the subset of T ∗ (G × G × G) obtained from the
T ∗ G −→
conormal bundle ν ∗ Λ by multiplying the cotangent vectors in the last factor by −1.
−→ T P is T Λ. Thus,
Also, it is clear that the graph of the tangent groupoid T G −→
according to Lemma 8.2, π # is a groupoid morphism if and only if (π # × π # ×
π # )(ν̄ ∗ Λ) ⊆ T Λ. The latter is clearly equivalent to saying that Λ is a coisotropic
submanifold of G × G × G (see [24]), or that G is a Poisson groupoid, by definition.
We can now prove that Lie bialgebroids do arise as infinitesimal invariants of
Poisson groupoids.
−→ P is a Poisson groupoid, then (AG, A∗ G) is a Lie bialgeTheorem 8.3. If G −→
broid.
#
Proof. Since πG
: T ∗ G −→ T G is a morphism of Lie groupoids, we can apply the
#
Lie functor and obtain A(πG
) : AT ∗ G −→ AT G, a morphism of Lie algebroids over
∗
a∗ : A G −→ T P .
We next prove that
∗
AT G
#
)
A(πG
−−−−−−−−−−−→
AT G
x








y
0
jG
T ∗ AG
−−−−−−−−−−−→
#
πAG
jG
(48)
T AG
0 −1
0
) : T ∗ AG −→ T ∗ T G, where ιT ∗ G : AT ∗ G −→
◦ ιT ∗ G ◦ (jG
commutes. Let λ = JG
∗
T T G is the inclusion. It is clear that λ is an injective morphism of double vector
bundles over ιG : AG −→ T G and e
1 : A∗ G −→ T ∗ G. Now πT#G ◦ λ = T (ιG ) ◦
#
πAG : T ∗ AG −→ T T G, and recalling that ιT G ◦ jG = JG ◦ T (ιG ), the commutativity
follows.
From Theorem 7.3 it now follows that
#
A(πG
)
∗
AT G −−−−−−−−−−−→ AT G
s
x




T ∗ A∗ G
x




−−−−−−−−−−−→
#
πAG
◦R
jG
T AG
commutes. We know that s is an isomorphism of Lie algebroids over A∗ G, and
from Theorem 7.1 we know that jG is an isomorphism of Lie algebroids over T P .
32
K. C. H. MACKENZIE AND PING XU
#
It follows that πAG
◦ R is a morphism of Lie algebroids over a∗ : A∗ G −→ T P and
by Theorem 6.2, this completes the proof.
Note that the commutativity of (48) shows that the construction of the Poisson
structure on the Lie algebroid of a Poisson groupoid generalizes the construction of
the tangent Poisson structure on the tangent bundle of a Poisson manifold.
Proposition 8.4. Let µ : G −→ G0 be a morphism of Poisson groupoids over
f : P −→ P 0 . Given φ0 ∈ ΓA∗ G0 denote by µ# (φ0 ) the section of A∗ G such that
lµ# (φ0 ) = lφ0 ◦ A(µ). Then
(1) [µ# (φ0 ), µ# (ψ 0 )] = µ# ([φ0 , ψ 0 ]) for all φ0 , ψ 0 ∈ ΓA∗ G0 ;
(2) T (f )! ◦ a∗ ◦ A∗ (µ)! = f ! (a0∗ ), where T (f )! : T P −→ f ! T P 0 is the induced map
over P , A∗ (µ)! : f ! A∗ G0 −→ A∗ G is the dual of the induced map A(µ)! over P , and
f ! (a0∗ ) : f ! A∗ G0 −→ f ! T P 0 is the pullback of a0∗ .
The proof is straightforward. One may define a morphism of Lie bialgebroids
(A, A∗ ) −→ (B, B ∗ ) to be a morphism of Lie algebroids µ : A −→ B over f : P −→
Q such that conditions (1) and (2) above are satisfied. In the terminology of [10],
a morphism of Lie bialgebroids is a morphism of Lie algebroids whose dual is a
comorphism of Lie algebroids.
It would now be possible to extend to Poisson groupoids many of the technical
results known for Poisson Lie groups (for example, [16]). However, we prefer to
postpone this to another occasion.
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Department of Pure Mathematics, University of Sheffield, Sheffield, S3 7RH, England
E-mail address: K.Mackenzie@uk.ac.sheffield
Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720,
USA. Permanent address: Department of Mathematics, University of Pennsylvania,
Philadelphia, PA 19104, USA.
E-mail address: ping@msri.org