Normal forms for Poisson maps and symplectic groupoids around Poisson transversals
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Abstract
Poisson transversals are submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In this communication, we prove a normal form theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous normal forms exist around such transversals, for which the Poisson map becomes transversally linear, and intertwines the normal form data of the transversals. Our second result concerns symplectic integrations. We prove that a neighborhood of a Poisson transversal is integrable exactly when the Poisson transversal itself is integrable, and in that case we prove a normal form theorem for the symplectic groupoid around its restriction to the Poisson transversal, which puts all structure maps in normal form. We conclude by illustrating our results with examples arising from Lie algebras.
Keywords
Symplectic manifolds, general Poisson manifolds Poisson groupoids and algebroids Lie algebras and Lie superalgebrasMathematics Subject Classification
53D05 53D17 17Bxx1 Introduction
Poisson transversals are special submanifolds which play in Poisson Geometry a role similar to that of symplectic submanifolds in Symplectic Geometry, and complete transversals in Foliation Theory. A Poisson transversal of a Poisson manifold \((M,\pi )\) is an embedded submanifold \(X\subset M\) which intersects all symplectic leaves transversally and symplectically. These submanifolds lie at the heart of Poisson geometry, appearing in many constructions and arguments already since the foundational work of Alan Weinstein [22].
In this communication, we continue our analysis of local properties around Poisson transversals with normal form results for Poisson maps and symplectic groupoids.
That Poisson transversals behave functorially with respect to Poisson maps has already been pointed out in [10]: a Poisson map pulls back Poisson transversals to Poisson transversals, and in fact, it pulls back the corresponding infinitesimal data pertaining to their normal forms. We prove that the two Poisson structures and the Poisson map can be put in normal form simultaneously:
Theorem 1
Let us remark that there are relatively few normal form results for Poisson maps in the literature (mostly for moment maps on symplectic manifolds [11, 15] and, in particular, for integrable systems [7, 8, 18]), and that our result is of a very different nature: it holds without any “compactness/properness assumptions”, which is just further evidence of the central role played by Poisson transversals in Poisson geometry.^{1}
Next, we move to symplectic groupoids. As a general principle, which follows from the normal form theorem, Poisson transversals encode all the geometry of a neighborhood in the ambient manifold, and ’transverse properties’ should hold for the transversal if and only if they hold true around it. We show that integrability by a symplectic groupoid is one such transverse property:
Theorem 2
(Integrability as a transverse property) A Poisson transversal is integrable if and only if it has an integrable open neighborhood.
In fact, we show much more:
Theorem 3
 (a)A symplectic groupoid integrating \(\pi _X^{\sigma }\) is \((\mathcal {G}_X^E,\omega _E) \rightrightarrows (E,\pi _X^{\sigma })\), where:Here \(\mathcal {P}(\mathcal {M}) \rightrightarrows \mathcal {M}\) stands for the pair groupoid of a manifold \(\mathcal {M}\), and \(\mathbf {p}:\mathcal {G}_X^E \rightarrow \mathcal {G}_X\) stands for the canonical groupoid map.$$\begin{aligned} \mathcal {G}_X^E:=\mathcal {G}_X \times _{\mathcal {P}(X)}\mathcal {P}(E), \ \ \ \ \omega _E:=\mathbf {p}^{*}(\omega _{X})+\mathbf {s}^{*}(\sigma )\mathbf {t}^{*}(\sigma ). \end{aligned}$$
 (b)
The restriction to E of any symplectic groupoid \((\mathcal {G},\omega _{\mathcal {G}}) \rightrightarrows (M,\pi )\) integrating \(\pi \) is isomorphic to the model \((\mathcal {G}_X^E,\omega _E)\) corresponding to \(\mathcal {G}_X:=\mathcal {G}_X\), \(\omega _X:=\omega _{\mathcal {G}}_{\mathcal {G}_X}\).
We conclude the paper by illustrating our results in the setting of linear Poisson structures, i.e., Lie algebras. Although in this linear setting the conclusions of our results are wellknown, the usage of Poisson transversals gives a new perspective on some classical results.
2 Preliminaries on Poisson transversals
Definition 1
 1.
\(m_t^{*}(\mathcal {V})=t\mathcal {V}\), for all \(t>0\);
 2.
\({\text {pr}}_{*}\mathcal {V}(\xi )=\pi ^{\sharp }(\xi )\), for all \(\xi \in T^{*}M\),
Remark 1
A spray on a Poisson manifold \((M,\pi )\) can be easily constructed: e.g., the horizontal lift \(\mathcal {V}(\xi )\) of \(\pi ^{\sharp }(\xi )\) with respect to a fixed linear connection is a Poisson spray. More generally, a \(T^*M\) Lie algebroid connection \(\nabla \) on \(T^*M\) [9] induces a \(T^*M\)geodesic flow which comes from a spray \(\mathcal {V}_{\nabla }\). In fact, it is easily seen that any spray comes from a \(T^*M\)connection, and moreover, there is a unique such connection which is torsionfree. Thus, all our constructions can be done in terms of connection; however, we prefer the spray terminology.
The following result played a crucial role in the proof of the normal form theorem in [10]:
Theorem A
 1.
\(\phi \) is defined on \({\varSigma }_{\mathcal {V}} \times [0,1]\);
 2.
The closed twoform \({\varOmega }_{\mathcal {V}}:=\int _0^1\phi _t^{*}\omega _{\mathrm {can}}dt\) is symplectic on \({\varSigma }_{\mathcal {V}}\);
 3.The submersionsgive a full dual pair, where \(\exp _{\mathcal {V}}:={\text {pr}}\circ \phi _1\).$$\begin{aligned} (M,\pi ){\mathop {\longleftarrow }\limits ^{{\text {pr}}}}({\varSigma }_{\mathcal {V}},{\varOmega }_{\mathcal {V}}){\mathop {\longrightarrow }\limits ^{\exp _{\mathcal {V}}}}(M,\pi ) \end{aligned}$$
Let \(X\subset (M,\pi )\) be a Poisson transversal with associated pair \((\pi _X,w_X)\). We denote by \({\varUpsilon }(w_X)\) the space of all closed twoforms \(\sigma \in {\varOmega }^2(N^{*}X)\) which along X satisfy \(\sigma _X=w_X\in {\varGamma }(\bigwedge ^2 NX)\), where we identify \(\bigwedge ^2 NX\) with the space of vertical twoforms in \(\bigwedge ^2 T^*(N^*X)_X\). To each \(\sigma \in {\varUpsilon }(w_X)\), there corresponds a local model of \(\pi \) around X, which, in Diracgeometric terms, is described as the Poisson structure \(\pi _X^{\sigma }\) corresponding to the Dirac structure \({\text {pr}}^*(L_{\pi _X})^{\sigma }\). As shown in [10], \(\pi _X^{\sigma }\) is defined in a neighborhood of X in \(N^*X\), and for any other \(\sigma ' \in {\varUpsilon }(w_X)\), \(\pi _X^{\sigma }\) and \(\pi _X^{\sigma '}\) are Poisson diffeomorphic around X, by a diffeomorphism that fixes X to first order.
Theorem B
Remark 2
In Theorem B, \(\exp _{\mathcal {V}}\), and \(\omega _{\mathcal {V}}\) are defined only on small enough neighborhoods of X in \(N^*X\), but we still write \(\exp _{\mathcal {V}}:N^{*}X\rightarrow M\), and \(\omega _{\mathcal {V}}\in {\varUpsilon }(w_X)\). This convention will be used throughout Sect. 3, also for other maps and tensors, as it simplifies notation considerably.
3 Normal form for Poisson maps
The result below is a the first indication for a normal form theorem for Poisson maps should hold around Poisson transversals; we refer the reader to [10] for a proof:
Lemma 1
 1.
\(\varphi \) is transverse to \(X_1\);
 2.
\(X_0:=\varphi ^{1}(X_1)\) is also a Poisson transversal;
 3.
\(\varphi \) restricts to a Poisson map \(\varphi _{X_0}:(X_0,\pi _{X_0}) \rightarrow (X_1,\pi _{X_1})\);
 4.
The differential of \(\varphi \) along \(X_0\) restricts to a fiberwise linear isomorphism between embedded normal bundles \(\varphi _*_{NX_0}:NX_0\rightarrow NX_1\);
 5.The map \(F:N^*X_0\rightarrow N^*X_1\), \(F(\xi )=(\varphi ^*)^{1}(\xi )\), \(\xi \in N^*X_0\) is a fiberwise linear symplectomorphism between the symplectic vector bundles$$\begin{aligned}F:(N^*X_0,w_{X_0})\rightarrow (N^*X_1,w_{X_1}).\end{aligned}$$
We are ready to state the main result of this section. Consider the same setting as in Lemma 1.
Theorem 4
In other words, the theorem allows us to bring simultaneously both Poisson structures in normal form and the Poisson map becomes linear in the normal directions. This specializes to the normal form theorem of [10] by taking \(M_0=M_1\), \(X_0=X_1\) and \(\varphi ={\text {id}}\). Remark 2 applies also here: the result is only local around \(X_0\) and \(X_1\), as the exponential maps \(\exp _{\mathcal {V}_i}\) are defined only around \(X_i\). Moreover, if \(X_i\) are not closed submanifolds, then we can only guarantee that the sprays \(\mathcal {V}_i\) are defined around \(X_i\).
Proof (of Theorem 4)
We split the proof into five steps:
Applying this construction, find sprays \(\widetilde{\mathcal {V}}_i\) on \(T^*U_i\), for \(i=0,1\), which are tangent to \(C^*_i\), and extend \(\mathcal {V}_i\). Note that, if \(X_i\) is a closed submanifold of \(M_i\), then \(\widetilde{\mathcal {V}}_i\) can be extended to the entire \(T^*M_i\).
To simplify notation, we will denote \(\widetilde{\mathcal {V}}_i\) also by \(\mathcal {V}_i\).
Step 5: Compatibility of the twoforms. As in Theorems A and B, we denote by \({\varOmega }_{\mathcal {V}_i}:=\int _0^1({\varPhi }^t_{\mathcal {V}_i})^{*}\omega _{\mathrm {can}}dt\) and \(\omega _{\mathcal {V}_i}:={\varOmega }_{\mathcal {V}_i}_{N^*X_i}\). By Theorem B, the exponentials \(\exp _{\mathcal {V}_i}:(N^*X_i,\pi _{X_i}^{\omega _{\mathcal {V}_i}})\hookrightarrow (M_i,\pi _i)\) are Poisson diffeomorphisms around \(X_i\). Hence, also \(F:(N^*X_0,\pi _{X_0}^{\omega _{\mathcal {V}_0}})\rightarrow (N^*X_1,\pi _{X_1}^{\omega _{\mathcal {V}_1}})\) is a Poisson map in a neighborhood of \(X_0\). This does not directly imply that \(F^*(\omega _{\mathcal {V}_1})=\omega _{\mathcal {V}_0}\), and this is what we prove next.
4 Integrability
Symplectic groupoids are the natural objects integrating Poisson manifolds. In this section, we discuss the relation between integrability of a Poisson manifold and integrability of one of its transversals. For integrable Poisson manifolds, we give a normal form theorem for the symplectic groupoid around its restriction to a Poisson transversal.
4.1 Symplectic groupoids
We recollect here a few facts about symplectic groupoids and integrability of Poisson manifolds. For references, see [3, 4].
We denote the source/target maps of a Lie groupoid \(\mathcal {G}\rightrightarrows M\) by \(\mathbf {s},\mathbf {t}:\mathcal {G}\rightarrow M\), and the multiplication by \(\mathbf {m}:\mathcal {G}\times _{\mathbf {s},\mathbf {t}}\mathcal {G}\rightarrow \mathcal {G}\).
A Poisson manifold \((M,\pi )\) is called integrable if such a symplectic groupoid \((\mathcal {G},\omega )\) exists giving rise to \(\pi \), in which case the groupoid is said to integrate \((M,\pi )\).
Theorem 5
A Poisson transversal \((X,\pi _X)\) of a Poisson manifold \((M,\pi )\) is integrable if and only if the restriction \((U,\pi _U)\) of \(\pi \) to an open neighborhood U of X is an integrable Poisson manifold.
Proof
Step 1: If. Let \(({\varSigma },{\varOmega })\rightrightarrows (U,\pi )\) be a symplectic groupoid, and \(p:U \supset E\rightarrow X\) be a tubular neighborhood on which the normal form holds: \(\pi _E=\pi _X^{\sigma }\), for some closed twoform \(\sigma \) on E, satisfying \(\sigma (v)=0\) for all \(v\in TX\). Denote \(\mathcal {G}_X:={\varSigma }_X\), \(\omega _X:={\varOmega }_{\mathcal {G}_X}\). Then \(\pi _X\) is integrable by the symplectic groupoid \((\mathcal {G}_X,\omega _X)\rightrightarrows (X,\pi _X)\). This is proved in [4], in the more general setting of “LieDirac submanifolds” (Theorem 9); for completeness, we include a simple proof:
Step 2 : Only if. Recall [14] that integrability of a Poisson manifold by a symplectic groupoid is equivalent to integrability of its cotangent Lie algebroid. In particular, \(\mathcal {G}_X\) integrates \(T^*X\). By Theorem B and Lemma 2 below, in a tubular neighborhood \(p:E\rightarrow X\) of the Poisson transversal \((X,\pi _X)\subset (M,\pi )\), the cotangent Lie algebroid \(T^*E\) of \(\pi _E\) is isomorphic to the pullback Lie algebroid \(TE\times _{TX}T^*X\) of the cotangent Lie algebroid \(T^*X\) of \(\pi _X\) by p. By Proposition 1.3 [13], the pullback Lie algebroid \(TE\times _{TX}T^*X\) is integrable by the pullback groupoid (see below), and so \((E,\pi _X^{\sigma })\) is integrable. \(\square \)
An inconvenient feature of both Theorem 5 and its proof is that we are left with a poor understanding of how the symplectic groupoids integrating \((X,\pi _X)\) and a neighborhood of it are related. This is the issue we address in the next section.
5 Normal form for symplectic groupoids
Our next goal is to state and prove Theorem 6 below, which refines Theorem 5 in that it gives a precise description of the symplectic groupoid integrating a neighborhood of a Poisson transversal in terms of the symplectic groupoid integrating the Poisson transversal itself.

\((X,\pi _X)\) is a Poisson manifold;

\(p:E\rightarrow X\) is a surjective submersion;

\(\sigma \) is a closed twoform on E such that the Dirac structure \(p^*(L_{\pi _X})^{\sigma }\) corresponds to a globally defined Poisson structure \(\pi _X^{\sigma }\) on E.
Lemma 2
Proof
We present next a general construction for symplectic groupoids, which provides the local model of a symplectic groupoid around its restriction to a Poisson transversal.
5.1 A pullback construction for symplectic groupoids
The construction of the pullback groupoid is rather standard (according to [13], it dates back to Ehresmann). We reexamine the construction in the setting of symplectic groupoids, in order to obtain a more explicit proof of Theorem 5.
Proposition 1
\((\mathcal {G}_X^E,\omega _{E})\rightrightarrows (E,\pi _X^{\sigma })\) is a symplectic groupoid.
The proof of Proposition 1 uses some general remarks about Dirac structures and Dirac maps:
Lemma 3
 (a)
If k and i are backward Dirac maps, and l is forward Dirac, then j is also forward Dirac.
 (b)
If \(j:(A,L_A)\rightarrow (B,L_B)\) is forward Dirac, and \(\omega \) is a closed twoform on B, then j is also a forward Dirac map between the gaugetransformed Dirac structures: \(j:(A,L_A^{j^*(\omega )})\rightarrow (B,L_B^{\omega })\).
 (c)
If \(L_A\) is the graph of a closed twoform \(\omega \) on A, and \(L_B\) is the graph of a Poisson structure \(\pi \) on B, and \(j:(A,L_{A})\rightarrow (B,L_{B})\) is forward Dirac, then \(\ker (\omega )\subset \ker (j_{*})\).
Proof
(a) Observe that, counting dimensions, it suffices to show that \(j_{*}L_{A} \subset L_B\). Fix then \(a \in A\), and set \(b:=j(a)\), \(c:=i(a)\) and \(d:=k(b)=l(c)\). To further simplify the notation, we also let \(L_a:=L_{A,a}\), \(L_b:=L_{B,b}\), \(L_c:=L_{C,c}\), and \(L_d:=L_{D,d}\).
Choose \(X_B+\eta _B \in j_*(L_{a})\). This means that \(X_B=j_*(X_A)\), for some vector \(X_A\) with \(X_A + j^*(\eta _B)\in L_{a}\). Since i is a backward Dirac map, there is a covector \(\eta _C\) such that \(j^*(\eta _B)=i^*(\eta _C)\) and \(i_*(X_A)+\eta _C\in L_c\). Since \(i^*(\eta _C)=j^*(\eta _B)\), the dual of the pullback property for TA implies that there is a covector \(\eta _D\in T^*_dD\), with \(\eta _C=l^*(\eta _D)\) and \(\eta _B=k^*(\eta _D)\). Since l is a forward Dirac map, we have that \(l_*(X_C)+\eta _D \in L_d\). Commutativity of the diagram implies that \(l_*(X_C)=k_*(X_B)\). Thus \(k_*(X_B)+\eta _D \in L_d\), and \(k^*(\eta _D)=\eta _B\). Finally, since k is a backward Dirac map, \(X_B + \eta _B \in L_b\). Hence \(X_B+\eta _B \in L_b\), and the conclusion follows.
(b) Note that, again by dimensional reasons, we need only show that \(L_b^{\omega }\subset j_*(L_a^{j^*(\omega )})\). Choose \(a\in A\) and set \(b:=j(a)\), \(L_a:=L_{A,a}\) and \(L_B:=L_{B,b}\). Consider \(X_B + \eta _B \in L_b^{\omega }\). This means that \(X_B + \eta _B\iota _{X_B}\omega \in L_b\). Since j is a forward Dirac map, there is a vector \(X_A\) with \(X_B=j_*(X_A)\) and \(X_A + j^*(\eta _B\iota _{X_B}\omega )\in L_a\). Clearly, \(j^*(\iota _{X_B}\omega )=j^*(\iota _{j_*(X_A)}\omega )=\iota _{X_A}j^*(\omega )\). Hence \(X_A + j^*(\eta _B)\iota _{X_A}j^*(\omega ) \in L_a\), and so \(X_A+ j^*(\eta _B) \in L_a^{j^*(\omega )}\). This shows that \(X_B + \eta _B \in j_*(L_a^{j^*(\omega )})\).
(c) If \(V\in \ker (\omega )\), then \(V \in L_{\omega }\). But j forward Dirac implies \(j_*(V) \in L_{\pi }\), and therefore \(j_*(V)=0\). \(\square \)
Proof (of Proposition 1)
5.2 The normal form theorem
We are now ready to prove that the structure of a symplectic groupoid around a Poisson transversal is described by the pullback construction:
Theorem 6
Proof
We split the proof into three steps: constructing \({\varPsi }\) as an isomorphism of Lie groupoids, showing that it is a symplectomorphism, and finally, that it integrates the isomorphism of Lie algebroids \(TE\times _{TX}T^*X\cong T^*E\).
Step 1: Construction of Lie groupoid isomorphism \({\varPsi }\).
That \(\omega _E_{\mathcal {G}_X}=\widetilde{\omega }_E_{\mathcal {G}_X}\) follows by our construction: indeed, we have \({\varPsi }_{\mathcal {G}_X}=\text {id}\) and \(\omega _X={\varOmega }_E_{\mathcal {G}_X}\); since \(\sigma _X=0\), also \(\omega _X=\omega _E_{\mathcal {G}_X}\), and hence our conclusion.
6 Linear Poisson structures
In this section, we write our results explicitly for linear Poisson structures. Our goal is to illustrate Theorems A, B, 1 and 3 in this context, thus recasting and reproving some wellknown results in what (we would argue) is their proper setting.
Illustration 1
 (a)The Poisson manifold \((\mathfrak {g}^*,\pi _{\mathfrak {g}})\) carries a canonical, complete Poisson spray \(\mathcal {V}_{\mathfrak {g}}\), whose flow (under the identification \(T^*\mathfrak {g}^*=\mathfrak {g}\times \mathfrak {g}^*\)) is given by:$$\begin{aligned} \phi _t:T^*\mathfrak {g}^*\longrightarrow T^*\mathfrak {g}^*,\ \ (x,\xi )\mapsto (x,e^{t\mathrm {ad}_x^*}\xi ). \end{aligned}$$
 (b)Let \(\mathcal {O}(\mathfrak {g}) \subset \mathfrak {g}\) be the subspace where the Lietheoretic exponential map \(\exp :\mathfrak {g} \rightarrow G\) is a local diffeomorphism. Then the closed twoform:is symplectic exactly on \(\mathcal {O}(\mathfrak {g}) \times \mathfrak {g}^* \subset T^*\mathfrak {g}^*\), and gives rise to the full dual pair: Explicitly:$$\begin{aligned} {\varOmega }_{\mathfrak {g}}:=\int _0^1\phi _t^*\omega _{\mathrm {can}}\mathrm {d}t \ \ \in \ \ {\varOmega }^2(T^{*}\mathfrak {g}^{*}) \end{aligned}$$where \({\varXi }_{x_0}\) is the linear endomorphism of \(\mathfrak {g}\) given by:$$\begin{aligned} {\varOmega }_{\mathfrak {g}}((x,\xi ),(y,\eta ))_{(x_0,\xi _0)}=\xi ({\varXi }_{x_0}y)\eta ({\varXi }_{x_0}x)+\xi _0\left( \left[ {\varXi }_{x_0}x,{\varXi }_{x_0}y\right] \right) , \end{aligned}$$(10)$$\begin{aligned} {\varXi }_{x_0}(x)=\int _{0}^1e^{t\mathrm {ad}_{x_0}}(x)\mathrm {d}t=\frac{e^{\mathrm {ad}_{x_0}}\mathrm {Id}_{\mathfrak {g}}}{\mathrm {ad}_{x_0}}(x). \end{aligned}$$
 (c)\(X\subset \mathfrak {g}^*\) is a Poisson transversal if and only if, for every \(\lambda \in X\), the twoform \(\lambda \circ [\cdot ,\cdot ]\) is nondegenerate on the annihilator of \(T_{\lambda }X\). Moreover, under the identification \(N^*X=\bigcup _{\lambda \in X} N_{\lambda }^{*}X\times \{\lambda \}\subset \mathfrak {g}\times X\), we have a Poisson diffeomorphism in a neighborhood of X given by:$$\begin{aligned} \exp _{\mathcal {V}_{\mathfrak {g}}}:(N^*X,\pi _X^{{\varOmega }_{\mathfrak {g}}_{N^*X}})\rightarrow (\mathfrak {g}^*,\pi _{\mathfrak {g}}) \ \ (x,\lambda )\mapsto e^{\mathrm {ad}_x^*}\lambda ; \end{aligned}$$
 (d)If \(f:\mathfrak {g} \rightarrow \mathfrak {h}\) is a Lie algebra map, and \(Y \subset \mathfrak {h}^*\) is a Poisson transversal, then \(X:=(f^*)^{1}Y \subset \mathfrak {g}^*\) is a Poisson transversal, and f induces a bundle map F fitting into the commutative diagram of Poisson maps:
Proof (of Illustration 1)
 (a)The flow \(\phi _t\) in the statement has infinitesimal generator the vector field \(\mathcal {V}_{\mathfrak {g}}\in \mathfrak {X}^1(\mathfrak {g}\times \mathfrak {g}^*)\) given by:where \(\mathrm {ad}_xy=[x,y]\) since we use right invariant vector fields to define the Lie bracket, and this is clearly a spray.$$\begin{aligned} \mathcal {V}_{\mathfrak {g},(x,\xi )}:=(0,\mathrm {ad}_x^*\xi )=(0,\pi _{\mathfrak {g},\xi }^{\sharp }x)\in \mathfrak {g}\times \mathfrak {g}^*=T_{x}\mathfrak {g}\times T_{\xi }\mathfrak {g}^*, \end{aligned}$$
 (b)Since trajectories \(\phi _t(x,\xi )\) of \(\mathcal {V}_{\mathfrak {g}}\) are cotangent paths, they can be integrated to elements in the Lie groupoid, yielding a groupoid exponential map:where \(\exp :\mathfrak {g} \rightarrow G\) denotes the Lietheoretic exponential map.$$\begin{aligned} \mathrm {Exp}_{\mathcal {V}_{\mathfrak {g}}}:T^*\mathfrak {g}^*\longrightarrow G\ltimes \mathfrak {g}^*, \ \ (x,\xi )\mapsto (\exp (x),\xi ), \end{aligned}$$On the other hand, the spray exponential map \(\exp _{\mathcal {V}_{\mathfrak {g}}}\), i.e., the composition of \(\phi _1\) with the bundle projection \(T^*\mathfrak {g}^* \rightarrow \mathfrak {g}^*\), becomes \(\mathrm {Exp}_{\mathcal {V}_{\mathfrak {g}}}\) composed with the target map:Now, the pullback by \(\mathrm {Exp}_{\mathcal {V}_{\mathfrak {g}}}\) of the symplectic structure \({\varOmega }_G\) of (9) is given by the formula in Theorem A (see [4] for details); hence, the general considerations above imply that the twoform \({\varOmega }_{\mathfrak {g}}\) is given by:$$\begin{aligned} \exp _{\mathcal {V}_{\mathfrak {g}}}(x,\xi )=e^{\mathrm {ad}_x^*}\xi . \end{aligned}$$This implies that \({\varOmega }_{\mathfrak {g}}\) is nondegenerate exactly on \(\mathcal {O}(\mathfrak {g})\times \mathfrak {g}^*\), and that the following is a commutative diagram of Poisson maps: The explicit formula (10) for \({\varOmega }_{\mathfrak {g}}\) is obtained by pulling back \({\varOmega }_G\) from (9), and we conclude with the observation that the linear endomorphism \({\varXi }_{x_0}:\mathfrak {g} \rightarrow \mathfrak {g}\) is the left translation of the differential of \(\exp :\mathfrak {g}\rightarrow G\) at \(x_0\), and therefore it is invertible precisely on \(\mathcal {O}(\mathfrak {g})\).$$\begin{aligned} {\varOmega }_{\mathfrak {g}}=(\mathrm {Exp}_{\mathcal {V}_{\mathfrak {g}}})^*{\varOmega }_{G}. \end{aligned}$$(11)
 (c)Consider an affine subspace \(\lambda +L\) passing through a point \(\lambda \in \mathfrak {g}^*\) and with direction a linear subspace \(L\subset \mathfrak {g}^*\). Now, \(\lambda +L\) is a Poisson transversal in a neighborhood of \(\lambda \) if and only if the following condition is satisfied:equivalently:$$\begin{aligned} \mathfrak {g}^*=L\oplus L^{\circ }\cdot \lambda , \ \ L^{\circ }\cdot \lambda :=\{X\cdot \lambda : X\in L^{\circ }\}; \end{aligned}$$(13)The remaining claims are immediate.$$\begin{aligned} \lambda \circ [\cdot ,\cdot ]_{L^{\circ }\times L^{\circ }} \text { is a nondegenerate 2form on }L^{\circ }. \end{aligned}$$(14)
 (d)The dual map \(f^*:(\mathfrak {h}^*,\pi _{\mathfrak {h}})\longrightarrow (\mathfrak {g}^*,\pi _{\mathfrak {g}})\) to a Lie algebra map f is a Poisson map, hence by Lemma 1, \(f^*\) is transverse to X, \(Y:=(f^*)^{1}(X)\) is a Poisson transversal in \(\mathfrak {h}^*\), and \(f^*\) restricts to a Poisson mapMoreover, f restricts to a linear isomorphism between the conormal spaces Open image in new window , for all \(\mu \in Y\). The inverses can be put together in a vector bundle map Open image in new window covering \(f^*: Y \rightarrow X\), which is fiberwise a linear isomorphism.$$\begin{aligned} f^*_{Y}:(Y,\pi _{Y})\longrightarrow (X,\pi _{X}). \end{aligned}$$We conclude by showing that the diagram in the statement commutes. Let \((y,\xi )\in N^*Y\). Then \(F(y,\xi )=(x,f^*(\xi ))\in N^*X\), where x satisfies \(y=f(x)\). For any \(z\in \mathfrak {g}\), we have:where we have used that f is a Lie algebra map. Since \(f^*\) and the vertical maps are Poisson maps, it follows that also F is Poisson around Y.$$\begin{aligned}&\exp _{\mathcal {V}_{\mathfrak {g}}}\left( F(y,\xi )\right) (z)=\exp _{\mathcal {V}_{\mathfrak {g}}}((x,f^*\xi ))(z)=\left( e^{\mathrm {ad}_x^*}f^*\xi \right) (z)\\&\quad =\xi (f(e^{\mathrm {ad}_x}z))=\xi (e^{\mathrm {ad}_{f(x)}}f(z))=\xi (e^{\mathrm {ad}_{y}}f(z))\\&\quad =f^*(e^{\mathrm {ad}_y^*}\xi )(z)=f^*(\exp _{\mathcal {V}_{\mathfrak {g}}}(y,\xi ))(z), \end{aligned}$$
Our next illustration concerns the specialization of Theorem 6 for Poisson transversals complementary to coadjoint orbits, in the particularly convenient setting where the coadjoint action is proper at the orbit.
Illustration 2
 (a)Along \(\widetilde{X}:=\lambda +\mathfrak {g}_{\lambda }^*\), the Poisson tensor \(\pi _{\mathfrak {g}}\) decomposes as:where we identify \(\mathfrak {g}_{\lambda }^*=c^{\circ }\);$$\begin{aligned} (\lambda +\xi )\circ \pi _{\mathfrak {g}}=\xi \circ \pi _{\mathfrak {g}_{\lambda }}+(\lambda +\xi )\circ \pi _c\in \wedge ^2\mathfrak {g}_{\lambda }^*\oplus \wedge ^2c^* \end{aligned}$$(16)
 (b)
\(\widetilde{X}\) intersects all coadjoint orbits cleanly and symplectically, and hence inherits an induced Poisson structure \(\pi _{\widetilde{X}}\);
 (c)\(\pi _{\widetilde{X}}\) is globally linearizable through the Poisson isomorphism:
 (d)The subspace \(X \subset \widetilde{X}\) where \(\widetilde{X}\) is a Poisson transversal contains \(\lambda \), and for a product neighborhood of the origin \(V\times W\subset c\times \mathfrak {g}^*_{\lambda }\), the following map is an open Poisson embedding onto a neighborhood of \(\lambda \):where \(\sigma _{\lambda }\) is the pullback of \({\varOmega }_{\mathfrak {g}}\) via the map:$$\begin{aligned} \left( V\times W, \pi _{\mathfrak {g}_{\lambda }}^{\sigma _{\lambda }}\right) \hookrightarrow \left( \mathfrak {g}^*,\pi _{\mathfrak {g}}\right) , \ \ (x,\xi )\mapsto e^{\mathrm {ad}_x^*}(\lambda +\xi ), \end{aligned}$$(17)$$\begin{aligned} c\times \mathfrak {g}^*_{\lambda }\rightarrow \mathfrak {g}\times \mathfrak {g}^*, \ \ (x,\xi )\mapsto (x,\lambda +\xi ); \end{aligned}$$
 (e)If a Lie group G integrating \(\mathfrak {g}\) acts properly at \(\lambda \), and \(G_{\lambda }\) denotes the isotropy group at \(\lambda \), then, by shrinking \(W\subset \mathfrak {g}_{\lambda }^*\) if need be, the restriction of the symplectic groupoid \(G\ltimes \mathfrak {g}^*\) to the image of the map (17) is isomorphic to the product of the groupoid \(G_{\lambda }\ltimes W\rightrightarrows W\) with the pair groupoid \(V\times V\rightrightarrows V\), with symplectic structure:$$\begin{aligned}&\left( V\times \left( G_{\lambda }\ltimes W\right) \times V, \mathbf {s}^*(\sigma _{\lambda })+\mathbf {p}^*({\varOmega }_{G_{\lambda }})\mathbf {t}^*(\sigma _{\lambda })\right) \rightrightarrows \left( V\times W,\pi _{\mathfrak {g}_{\lambda }}^{\sigma _{\lambda }}\right) ,\ \ \text {where}\\&\quad \mathbf {s}(y,(g,\xi ),x)=(x,\xi ), \ \ \mathbf {p}(y,(g,\xi ),x)=(g,\xi ), \ \ \mathbf {t}(y,(g,\xi ),x)=(y,\mathrm {Ad}_{g^{1}}^*\xi ). \end{aligned}$$
Remark 3
It was first proved in [17] that the splitting condition (15) implies that the transverse Poisson structure to the coadjoint orbit at \(\lambda \) is linearizable, see also [23].
Submanifolds which intersect the symplectic leaves cleanly and symplectically, and for which the induced bivector is smooth, are called PoissonDirac [4]. In fact, the affine submanifold \(\lambda +\mathfrak {g}_{\lambda }^*\) turns out to be a LieDirac submanifold (also called “Dirac submanifold”), see [24, Example 2.18].
Proof (of Illustration 2)
As for (e), note that the properness assumption implies that the group \(G_{\lambda }\) is compact, that the coadjoint orbit through \(\lambda \) is closed, and that the splitting (15) can be assumed to be \(G_{\lambda }\)invariant. This assumption not only implies that the transverse Poisson structure is linearizable, but also that the Poisson manifold \((\mathfrak {g}^*,\pi _{\mathfrak {g}})\) is linearizable around the coadjoint orbit through \(\lambda \) in the sense of [21], see [6, Example 2.7].
A direct application of Theorem 6 now shows that the restriction of the symplectic groupoid \(G\ltimes \mathfrak {g}^*\) to the image of the map (17) is isomorphic to the symplectic groupoid from e). \(\square \)
Recall that a Lie algebra \(\mathfrak {h}\) is called a Frobenius Lie algebra if the coadjoint orbit through some \(\lambda \in \mathfrak {h}^*\) is open.
Illustration 3
 (a)There is an open neighborhood \(\mathcal {U}\) of 0 in \(\mathfrak {h}\) such that the twoform on \(\mathfrak {h}\)is nondegenerate on \(\mathcal {U}\), and the map:$$\begin{aligned} \omega _{\lambda ,x_0}(x,y)=\lambda \left( [{\varXi }_{x_0}x,{\varXi }_{x_0}y]\right) , \end{aligned}$$is a Poisson diffeomorphism onto a neighborhood of \(\lambda \) in \(\mathfrak {h}^*\);$$\begin{aligned} (\mathcal {U},\omega _{\lambda }^{1})\longrightarrow (\mathfrak {h}^*,\pi _{\mathfrak {h}}), \ \ x\mapsto e^{\mathrm {ad}_x^*}\lambda \end{aligned}$$(19)
 (b)Around the Poisson transversal \(X_{\lambda }:=r^{1}(\lambda )\) there is a global Weinstein splitting of \(\pi _{\mathfrak {g}}\) given by the commutative diagram of Poisson maps:
 (c)
Theorem 6 for \(X_{\lambda }\) implies that the restriction of the symplectic groupoid \((G\ltimes \mathfrak {g}^*,{\varOmega }_G)\) to the image of (20) is isomorphic to the product of the symplectic groupoid \((G\ltimes \mathfrak {g}^*,{\varOmega }_G)_{X_{\lambda }}\) and the symplectic pair groupoid \((\mathcal {U}\times \mathcal {U},\mathrm {pr}_1^*\omega _{\lambda }\mathrm {pr}_2^*\omega _{\lambda })\);
 (d)The Lietheoretic exponential of H, \(\exp :\mathfrak {h}\rightarrow H\), induces a factorization of diagram (20) through the commutative diagram of Poisson maps: where \(\widetilde{\lambda } \in {\varOmega }^1(H)\) is the leftinvariant oneform extending \(\lambda \), \(H_{\lambda }\) is the stabilizer of \(\lambda \), and the horizontal arrows are Hequivariant Poisson diffeomorphisms.
Part (d) gives a global description of the Poisson structure on the open \(\mathfrak {g}^*_S\), which implies the following:
Corollary 1
 (1)
The horizontal distribution is involutive, and is given by the tangent bundle to the Horbits;
 (2)
The horizontal twoform is the pullback of the symplectic form on the leaf S;
 (3)
In the decomposition \(\pi _{\mathfrak {g}}_{r^{1}(S)}=\pi ^{\mathrm {v}}+\pi ^{\mathrm {h}}\) into vertical and horizontal components, we have that both bivectors are Poisson and commute.
Remark 4
Note that, in general, the open set \(\mathfrak {g}^*_S\) is not saturated; for example, if \(\mathfrak {h}\) is the diagonal subalgebra in \(\mathfrak {g}=\mathfrak {aff}(1)\oplus \mathfrak {aff}(1)\).
We note also the following surprising property:
Corollary 2
The induced Poisson structure \(\pi _{X_{\mu }}\) on the Poisson transversal \(X_{\mu }\) is at most quadratic for the canonical \(\mathfrak {h}^{\circ }\)affine space structure on \(X_{\mu }\).
Remark 5
A special case of this corollary appeared in [19] when considering the transverse Poisson structure to the coadjoint orbit through an element \(\xi \in \mathfrak {g}^*\) for which the isotropy Lie algebra \(\mathfrak {g}_{\xi }\) has a complement \(\mathfrak {h}\) which is also a Lie algebra. In this case, note that by (14) \(\mathfrak {h}\) is a Frobenius algebra whose orbit through \(\lambda :=\xi _{\mathfrak {h}}\) is open, and \(X_{\lambda }:=\xi +\mathfrak {h}^{\circ }\) is a Poisson transversal to the coadjoint orbit of complementary dimension. Thus the corollary implies the main result of [19].
Proof
Note that \(\{\lambda \}\) is itself a Poisson transversal, with conormal bundle \(\mathfrak {h}\times \{\lambda \}\), and that the pullback of \({\varOmega }_{\mathfrak {h}}\) under \(i_{\mathfrak {h}\times \{\lambda \}}:\mathfrak {h}\rightarrow \mathfrak {h}\times \{\lambda \}\), \(x\mapsto (x,\lambda )\) is given by \(\omega _{\lambda }\). Thus, Theorem B specializes to the diffeomorphism claimed in (a).
Part (c) is a direct consequence of Theorem 6.
Proof (of Corollary 1)
By Lemma 1, each fiber \(X_{\mu }:=r^{1}(\mu )\), \(\mu \in S\), is a Poisson transversal; or equivalently, \(\pi _{\mathfrak {g}}\) is horizontally nondegenerate for the map (22).
Proof (of Corollary 2)
Footnotes
 1.
Following a suggestion of the referee, let us also point out, as a guide for the reader to Theorems 2 and 3, that in symplectic geometry one has tubular neighborhood theorems for symplectic and Lagrangian submanifolds. Now LieDirac submanifolds (of which Poisson transvsersals are examples) correspond to symplectic subgroupoids [4, Theorem 9], while the graph of a Poisson map \(\varphi : (M_0,\pi _0) \rightarrow (M_1,\pi _1)\), sitting as a coisotropic submanifold in \((M_0,\pi _0) \times (M_1,\pi _1)\), corresponds to a Lagrangian subgroupoid [2, Theorem 5.4].
 2.
Note that our sign convention is different from that in [1]; namely, the IMform corresponding to a closed twoform \(\eta \) on a groupoid \(\mathcal {G}\), is given by \(A\ni V\mapsto \mathbf {u}^*(\iota _V\eta )\), where \(\mathbf {u}:M\rightarrow \mathcal {G}\) is the unit map.
Notes
Acknowledgements
The first author was supported by the NWO Vrije Competitie project “Flexibility and Rigidity of Geometric Structures” No. 612.001.101 and by IMPA (CAPESFORTAL project) the second was supported by the NWO Veni Grant 613.009.031 and by the NSF grant DMS 1405671. We would like to thank Marius Crainic for the many good conversations, and for his insight that Lemma 1 should be a shadow of a normal form theorem for Poisson maps. We would also like to thank David MartínezTorres and Rui Loja Fernandes for useful discussions.
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