---
abstract: 'We study some properties of coisotropic submanifolds of a manifold with respect to a given multivector field. Using this notion, we generalize the results of Weinstein [@wein] from Poisson bivector field to Nambu-Poisson tensor or more generally to any multivector field. We also introduce the notion of Nambu-Lie groupoid generalizing the concepts of both Poisson-Lie groupoid and Nambu-Lie group. We show that the infinitesimal version of Nambu-Lie groupoid is the notion of weak Lie-Filippov bialgebroid as introduced in [@bas-bas-das-muk]. Next we introduce coisotropic subgroupoids of a Nambu-Lie groupoid and these subgroupoids corresponds to, so called coisotropic subalgebroids of the corresponding weak Lie-Filippov bialgebroid.'
address: 'Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, West Bengal, India.'
author:
- Apurba Das
title:
- 
- Multiplicative Nambu structures on Lie groupoids
---

Introduction
============

In [@wein], Weinstein introduced the notion of coisotropic submanifold of a Poisson manifold generalizing the notion of Lagrangian submanifold of symplectic manifold. A submanifold $C$ of a Poisson manifold $(P, \pi)$ is called coisotropic, if $\pi^{\sharp}(TC)^0 \subset TC,$ or, equivalently $\pi (\alpha, \beta) = 0$, for all $ \alpha, \beta \in (TC)^0$, where $(TC)^0$ is the conormal bundle of $C$. Moreover, Weinstein proved the following results.

1.  A map $\phi : P_1 \rightarrow P_2$ between Poisson manifolds is a Poisson map if and only if its graph is a coisotropic submanifold of $P_1 \times P_2^{-},$ where $P_2^{-}$ stands for the manifold $P_2$ with opposite Poisson structure.

2.  If $\phi : P \rightarrow Q$ is a surjective submersion from a Poisson manifold $P$ to some manifold $Q$, then $Q$ has a Poisson structure for which $\phi$ is a Poisson map if and only if $$\{(x,y)| \phi(x) = \phi(y)\} \subset P \times P$$ is a coisotropic submanifold of $P \times P^{-}.$

To define the coisotropic submanifold of a Poisson manifold, one does not require the Poisson tensor to be closed, that is, $[\pi, \pi] = 0,$ where $[~,~]$ denotes the Schouten bracket on multivector fields. Therefore, the notion of coisotropic submanifolds make sense for any bivector field, or more generally, for any multivector field. Explicitly, if $M$ is a smooth manifold and $\Pi \in \mathcal{X}^n(M) = \Gamma \bigwedge^n TM$ be an $n$-vector field on $M$, then a submanifold $C \hookrightarrow M$ is called coisotropic with respect to $\Pi$ if $$\Pi^{\sharp} ({\bigwedge}^{n-1}(TC)^0) \subset TC ~~\Leftrightarrow ~~\Pi (\alpha_1, \ldots, \alpha_n) = 0,~~ \mbox{for all}~~ \alpha_1, \ldots, \alpha_n \in (TC)^0,$$ where $\Pi^\sharp : \bigwedge^{n-1}T^*M \rightarrow TM$ is the bundle map induced by $\Pi$.

Nambu-Poisson manifolds are generalization of Poisson manifolds. Recall that a Nambu-Poisson manifold of order $n$ is a manifold $M$ equipped with an $n$-vector field $\Pi$ such that the induced bracket on functions satisfies the Fundamental identity (Definition \[nambu-poisson\]). The $n$-vector field $\Pi$ of a Nambu-Poisson manifold is referred to as the associated Nambu tensor. Coisotropic submanifolds of a Nambu-Poisson manifold $M$ are those submanifolds which are coisotropic with respect to the Nambu tensor $\Pi$.

In the present paper, we study some basic properties of coisotropic submanifolds of a manifold with respect to a given multivector field and generalize the results of Weinstein to the case of multivector field. More precisely, we prove the following results (Propositions \[nambu map-coiso\] and \[coinduced-coiso\]).

1.  Let $(M, \Pi_M)$ and $(N, \Pi_N)$ be two manifolds with $n$-vector fields and $\phi: M \rightarrow N$ be a smooth map. Then $\phi_* \Pi_M = \Pi_N$ if and only if its graph $$\text{Gr}(\phi) := \{(m, \phi(m))| m \in M\}$$ is a coisotropic submanifold of $M \times N$ with resoect to $\Pi_M \oplus (-1)^{n-1} \Pi_N$.

2.  Let $(M, \Pi_M)$ be a manifold with an $n$-vector field and $\phi: M \rightarrow N$ be a surjective submersion. Then $N$ has an (unique) $n$-vector field $\Pi_N$ such that $\phi_* \Pi_M = \Pi_N$ if and only if $$R(\phi) := \{(x,y) \in M \times M|\phi(x)=\phi(y)\}$$ is a coisotropic submanifold of $M \times M$ with respect to $\Pi_M \oplus (-1)^{n-1}{\Pi}_M$.

Poisson Lie group is a Lie group equipped with a Poisson structure such that the group multiplication map is a Poisson map. Equivalently, a Lie group equipped with a Poisson structure is a Poisson Lie group if the Poisson bivector field is multiplicative [@lu-wein]. These definitions have no natural extension when one wants to define Poisson groupoid. Nevertheless, the notion of coisotropic submanifolds of Poisson manifolds was used by Weinstein [@wein] to introduce the notion of Poisson groupoid. Recall that a Poisson groupoid is a Lie groupoid $G \rightrightarrows M$ with a Poisson structure on $G$ such that the graph of the groupoid (partial) multiplication map is a coisotropic submanifold of $G \times G \times G^{-}.$

In [@xu], P. Xu gave an equivalent formulation of Poisson groupoid which generalizes the multiplicativity condition for Poisson Lie group. More generally, In [@pont-geng-xu], the authors introduced the notion of multiplicative multivector fields on a Lie groupoid. Given a Lie groupoid $G \rightrightarrows M$, an $n$-vector field $\Pi \in \mathcal{X}^{n}(G)$ is called multiplicative, if the graph of the groupoid multiplication is a coisotropic submanifold of $G \times G \times G$ with respect to $\Pi \oplus \Pi \oplus (-1)^{n-1} \Pi$. In this terminology, a Poisson groupoid is a Lie groupoid equipped with a multiplicative Poisson tensor.

In the present paper, we extend this approach to the case of Lie groupoid with a Nambu structure. We introduce the notion of a Nambu-Lie groupoid as a Lie groupoid with a Nambu structure $\Pi$ such that the Nambu tensor $\Pi$ is multiplicative (Definition \[nambu-lie groupoid\]). When $G$ is a Lie group, this definition coincides with the definition of Nambu-Lie group given by Vaisman [@vais]. Using results proved in [@pont-geng-xu] for mutiplicative multivector fields on Lie groupoid, we deduce the following facts which are parallel to the case of Poisson groupoid. Suppose $(G \rightrightarrows M, \Pi)$ is a Nambu-Lie groupoid, then

1.  $M \hookrightarrow G$ is a coisotropic submanifold of $G$;

2.  the groupoid inversion map $i: G \rightarrow G$ is an anti Nambu-Poisson map;

3.  there is a unique Nambu-Poisson structure $\Pi_M$ on $M$ for which the source map is a Nambu-Poisson map (Proposition \[inverse-basenambu\]).

It is well known that for a Nambu-Poisson manifold $M$ of order $n$, the space of $1$-forms admits a skew-symmetric $n$-bracket which satisfies the Fundamental identity modulo some restriction ([@vais; @gra-mar; @bas-bas-das-muk]). Moreover, the bracket on forms and the de-Rham differential of the manifold satisfy a compatibility condition similar to that of a Lie bialgebroid. This motivates the authors [@bas-bas-das-muk] to introduce a notion of weak Lie-Filippov bialgebroid of order $n$. If $M$ is a Nambu-Poisson manifold of order $n$, then $(TM, T^*M)$ provides such an example. Roughly speaking, a weak Lie-Filippov bialgebroid of order $n$ ($n > 2$) over $M$ is a Lie algebroid $A \rightarrow M$ together with a skew-symmetric $n$-ary bracket on the space of sections of the dual bundle $A^* \rightarrow M$ and a bundle map $\rho : \bigwedge^{n-1}A^* \rightarrow TM$ satisfying some conditions (cf. Definition \[lie-fill-defn\]). Moreover it is proved in [@bas-bas-das-muk] that, if $(A, A^*)$ is a weak Lie-Filippov bialgebroid of order $n$ over $M$, then there is an induced Nambu-Poisson structure of order $n$ on the base manifold $M$.

In the present paper, we prove that weak Lie-Filippov bialgebroids are infinitesimal form of Nambu-Lie groupoids. Explicitly, if $C$ is a coisotropic submanifold of a Nambu-Poisson manifold $(M, \Pi)$, then we show that the $n$-ary bracket on the space of $1$-forms on $M$ restricts to the sections of the conormal bundle $(TC)^0 \rightarrow C$ and the induced bundle map $\Pi^{\sharp} : \bigwedge^{n-1}T^*M \rightarrow TM$ maps $\bigwedge^{n-1}(TC)^0$ to $TC$ (Proposition \[coiso-n-bracket\]). Therefore, if $G \rightrightarrows M$ is a Nambu-Lie groupoid of order $n$ whose Lie algebroid is $AG \rightarrow M,$ then as $M$ is a coisotropic submanifold of $G$, the space of sections of the dual bundle $A^*G \cong (TM)^0 \rightarrow M$ is equipped with a skew-symmetric $n$-ary bracket. Moreover, there is a bundle map $\bigwedge^{n-1} A^*G \rightarrow TM$ so that the pair $(AG, A^\ast G),$ with the above data, satisfies the conditions of a weak Lie-Filippov bialgebroid. Thus we prove the following (cf. Theorem \[nambu-grpd-bialgbd\]).

Let $(G \rightrightarrows M , \Pi) $ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$. Then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ over $M$.

Finally, we compare the Nambu-Poisson structures on the base manifold $M$ induced from the Nambu-Lie groupoid $G \rightrightarrows M$ and the weak Lie-Filippov bialgebroid $(AG, A^*G)$ (cf. Proposition \[compare-two-NP-structure\]).

Next, we introduce the notion of a coisotropic subgroupoid $H \rightrightarrows N$ of a Nambu-Lie groupoid $(G \rightrightarrows M, \Pi).$ To study the infinitesimal form of a coisotropic subgroupoid, we introduce the notion of coisotropic subalgebroid of a weak Lie-Filippov bialgebroid. Then we show that the Lie algebroid of a coisotropic subgroupoid $H \rightrightarrows N$ is a coisotropic subalgebroid of the corresponding weak Lie-Filippov bialgebroid $(AG, A^*G)$ (cf. Proposition \[coiso-subgrpd-coiso-subalgbd\]).

[**Organization.**]{} The paper is organized as follows. In section 2, we recall basic definitions and conventions. In section 3, we study some properties of coisotropic submanifolds of a manifold with respect to a given multivector field and in section 4 we introduce Nambu-Lie groupoids and study some of its basic properties. In section 5, we show that the infinitesimal object corresponding to Nambu-Lie groupoid is weak Lie-Filippov bialgebroid and in section 6 we introduce coisotropic subgroupoids of a Nambu-Lie groupoid and study their infinitesimal.\
[**Acknowledgements.**]{} The author would like to thank his Ph.D supervisor Professor Goutam Mukherjee for his guidance and carefully reading the manuscript.

Preliminaries
=============

In this section, we recall some basic preliminaries from [@duf-zung; @mac-book; @xu] and fix the notations that will be used throughout the paper .

Nambu-Poisson manifolds are $n$-ary generalizations of Poisson manifolds introduced by Takhtajan [@takhtajan].

\[nambu-poisson\] Let $M$ be a smooth manifold. A [*Nambu-Poisson structure*]{} of order $n$ on $M$ is a skew-symmetric $n$-multilinear bracket $$\begin{aligned}
 \{~, \ldots, ~\} : C^\infty (M) \times \stackrel{(n)}{\ldots} \times C^\infty (M) \rightarrow C^\infty (M)\end{aligned}$$ satisfying the following conditions:

1.  [*Leibniz rule*]{}:\
    $\{f_1, \ldots, f_{n-1}, fg\} = f \{f_1, \ldots, f_{n-1}, g \} + \{f_1, \ldots, f_{n-1}, f \} g; $

2.  [*Fundamental identity*]{}:\
    $\{f_1, \ldots ,f_{n-1}, \{g_1,\ldots, g_n\}\} = \sum_{k=1}^n  \{g_1,\ldots,g_{k-1},\{f_1,\ldots,f_{n-1},g_k\},\ldots, g_n\}; $

for all $f_i, g_j, f, g \in C^\infty (M).$ A manifold together with a Nambu-Poisson structure of order $n$ is called a [*Nambu-Poisson manifold of order $n$*]{}. Thus the space of smooth functions with this bracket forms a Nambu-Poisson algebra. A Nambu-Poisson manifold of order $2$ is nothing but a Poisson manifold [@vais-book]. Since the bracket above is skew-symmetric and satisfies Leibniz rule, there exists an $n$-vector field $\Pi \in \mathcal{X}^{n}(M)$, such that $$\{f_1, \ldots, f_n\} = \Pi (df_1, \ldots, df_n),$$ for all $ f_1, \ldots, f_n \in C^\infty (M)$. Given any $(n-1)$ functions $ f_1, \ldots, f_{n-1} \in C^\infty (M)$, the [*Hamiltonian vector field*]{} associated to these functions is denoted by $X_{f_1\ldots f_{n-1}}$ and is defined by $$X_{f_1\ldots f_{n-1}} (g) = \{f_1, \ldots ,f_{n-1}, g\}.$$ Note that the Fundamental identity, in terms of Hamiltonian vector fields, is equivalent to the condition $$[X_{f_1\ldots f_{n-1}}, X_{g_1\ldots g_{n-1}}] = \sum_{k=1}^{n-1} X_{g_1\ldots \{f_1, \ldots, f_{n-1},g_k\}\ldots g_{n-1}}.$$ for all $f_1, \ldots, f_{n-1}, g_1, \ldots, g_{n-1} \in C^\infty(M).$ The Fundamental identity can also be rephrased as $$\mathcal{L}_{X_{f_1\ldots f_{n-1}}} \Pi = 0,$$ for all $f_1, \ldots, f_{n-1} \in C^\infty(M),$ which shows that every Hamiltonian vector field preserves the Nambu tensor. A Nambu-Poisson manifold is often denoted by $(M, \{~, \ldots, ~\})$ or simply by $(M, \Pi).$

1.  Let $M$ be an orientable manifold of dimension $n$ and $\nu$ be a volume form on $M$. Define an $n$-bracket $\{~, \ldots, ~\}$ on $C^\infty(M)$ by the following identity $$df_1 \wedge \cdots \wedge df_n = \{f_1, \ldots, f_n\} \nu.$$ Then $\{~, \ldots, ~\}$ defines a Nambu-Poisson structure of order $n$ on $M$. Let $\Pi_\nu \in \Gamma{\bigwedge^n TM}$ denotes the associated Nambu-Poisson tensor. If $\Pi \in \Gamma{\bigwedge^n TM}$ is any Nambu-Poisson structure of order $n$ such that $\Pi \neq 0$ at every point, then there exists a volume form $\nu'$ on $M$ such that $\Pi = \Pi_{\nu'}$. If $M = \mathbb{R}^n$ and $\nu = dx_1 \wedge \cdots \wedge dx_n$ is the standard volume form, then one recovers the Nambu structure on $\mathbb{R}^n$ originally discussed by Y. Nambu [@nambu].

2.  Let $\Pi$ be any $n$-vector field on an oriented manifold $M$ of dimension $n$. Then $\Pi$ defines a Nambu structure of order $n$ on $M$ (see [@ibanez1]).

3.  Let $M$ be a manifold of dimension $m$ and $X_1, \ldots, X_n$ be linearly independent vector fields such that $[X_i, X_j] = 0$ for all $i, j = 1, \ldots, n$. Then the $n$-vector field $\Pi = X_1 \wedge \cdots \wedge X_n$ defines a Nambu structure of order $n$.

4.  Let $(M, \{~, \ldots, ~\})$ be a Nambu-Poisson manifold of order $n$. Suppose $k \leqslant n-2$ and $F_1, \ldots, F_k \in C^\infty(M)$ be any fixed functions on $M$. Define a $(n-k)$-bracket $\{~, \ldots, ~\}'$ on $C^\infty(M)$ by $$\begin{aligned}
     \{f_1, \ldots, f_{n-k}\}' = \{F_1, \ldots, F_k, f_1, \ldots, f_{n-k}\}\end{aligned}$$ for $f_1, \ldots, f_{n-k} \in C^\infty(M)$. Then $\{~, \ldots, ~\}'$ defines a Nambu-structure of order $(n-k)$ on $M$. This Nambu structure is called the [*subordinate Nambu structure*]{} of $(M, \{~, \ldots, ~\})$ with subordinate function $F_1, \ldots, F_k$.

5.  If $\Pi_i$ is a Nambu structure of order $n_i$ on a manifold $M_i$ $(i=1,2)$, then $\Pi = \Pi_1 \wedge \Pi_2$ is a Nambu structure of order $n_1 + n_2$ on $M_1 \times M_2$ [@duf-zung].

More examples of Nambu structures can be found in [@ibanez2; @vais].

Let $(M, \Pi)$ be a Nambu-Poisson manifold of order $n$. For each $m \in M$, let $\mathcal{D}_mM \subset T_mM$ be the subspace of the tangent space at $m$ generated by all Hamiltonian vector fields at $m$. Since the Lie bracket of two Hamiltonians is again a Hamiltonian, therefore $\mathcal{D}$ defines a [*(singular) integrable distribution*]{} whose leaves are either $n$-dimensional submanifolds endowed with a volume form or just singletons [@duf-zung].

Let $(M, \Pi_M)$ and $(N, \Pi_N)$ be two manifolds with $n$-vector fields. A smooth map $\phi: M \rightarrow N$ is called [*$(\Pi_M, \Pi_N)$-map*]{} if the induced brackets on functions satisfies: $$\begin{aligned}
 \{\phi^*f_1, \ldots, \phi^*f_n\}_M = \phi^* \{f_1, \ldots, f_n\}_N\end{aligned}$$ for all $f_1, \ldots, f_n \in C^\infty(N),$ or equivalently, $\phi_* \Pi_M = \Pi_N.$ The map $\phi$ is called an [*anti $(\Pi_M, \Pi_N)$-map*]{} if $$\begin{aligned}
 \{\phi^*f_1, \ldots, \phi^*f_n\}_M = (-1)^{n-1} \phi^* \{f_1, \ldots, f_n\}_N\end{aligned}$$ for all $f_1, \ldots, f_n \in C^\infty(N)$. A $(\Pi_M, \Pi_N)$-map $\phi : (M, \Pi_M) \longrightarrow (N, \Pi_N)$ between Nambu-Poisson manifolds of the same order $n$ is called a [*Nambu-Poisson map*]{} or a $N$-$P$-map.

The condition for a $(\Pi_M, \Pi_N)$-map can also be expressed in terms of the induced bundle maps as $$\begin{aligned}
\Pi^{\sharp}_{N, {\phi(m)}} = T_m \phi \circ \Pi^{\sharp}_{M, {m}} \circ T^*_m \phi ~ ~~\mbox{~~ for each}~~ m \in M,\end{aligned}$$ where $\Pi^{\sharp}_M :\bigwedge^{n-1}T^*M \rightarrow TM$ is the induced bundle map and is given by $$\begin{aligned}
 \langle \beta, \Pi^{\sharp}_M (\alpha_1 \wedge \cdots \wedge \alpha_{n-1}) \rangle = \Pi_M (\alpha_1, \ldots, \alpha_{n-1}, \beta)\end{aligned}$$ for all $\alpha_1, \ldots, \alpha_{n-1}, \beta \in T_x^* M,$ $x \in M$.

A [*Lie groupoid*]{} over a smooth manifold $M$ is a smooth manifold $G$ together with the following structure maps:

1.  two surjective submersions $\alpha, \beta : G \rightarrow M $, called the [*source*]{} map and the [*target*]{} map respectively;

2.  a smooth [*partial multiplication*]{} map $$G_{(2)} = \{(g,h) \in G \times G | \beta(g) = \alpha (h) \} \rightarrow G , ~~ (g,h)\mapsto gh ;$$

3.  a smooth [*unit*]{} map $\epsilon : M \rightarrow G$, $x \mapsto \epsilon_x ;$

4.  and a smooth [*inverse*]{} map $i : G \rightarrow G$, $g \mapsto g^{-1}$ with $\alpha (g^{-1}) = \beta (g)$ and $\beta (g^{-1}) = \alpha (g)$

such that, the following conditions are satisfied\
$\hspace*{1.5cm}$(i) $\alpha (gh) = \alpha (g)$ and $\beta(gh) = \beta (h)$;\
$\hspace*{1.5cm}$(ii) $(gh)k = g(hk),$ whenever the multiplications make sense;\
$\hspace*{1.5cm}$(iii) $\alpha(\epsilon_x) = \beta (\epsilon_x) = x$, $\forall x \in M$;\
$\hspace*{1.5cm}$(iv) $\epsilon_{\alpha(g)} g = g$ and $ g \epsilon_{\beta(g)} = g$, $\forall g \in G$;\
$\hspace*{1.5cm}$(v) $g g^{-1} = \epsilon_{\alpha(g)}$ and $g^{-1} g = \epsilon_{\beta(g)}$, $\forall g \in G$.

A Lie groupoid $G$ over $M$ is denoted by $G \rightrightarrows M$ when all the structure maps are understood.

Note that the smooth structure on $G_{(2)}$ comes from the fact that $$G_{(2)} = (\beta \times \alpha)^{-1}(\Delta_M),$$ where $\beta \times \alpha : G \times G \rightarrow M \times M,~~(g,h) \mapsto (\beta(g), \alpha (h))$ and $ \Delta_M = \{(m,m)| m \in M\} \subset M \times M$ is the diagonal submanifold of $M \times M.$ Then these conditions imply that the inverse map $i : G \rightarrow G,$ $g \mapsto g^{-1}$ is also smooth [@mac-book]. Moreover, $\alpha$-fibers and $\beta$-fibers are submanifolds of $G$ as both $\alpha$ and $\beta$ are surjective submersions.

Given a Lie groupoid $G \rightrightarrows M$, define an equivalence relation $'\sim'$ on $M$ by the following: two points $x, y \in M$ are said to be equivalent, written as $x \sim y$, if there exists an element $g \in G$ such that $\alpha(g) = x$, $\beta(g) = y$. The quotient $M/\sim$ is called the [*orbit set*]{} of $G$.

Given two Lie groupoid $G_1 \rightrightarrows M_1$ and $G_2 \rightrightarrows M_2$, a [*morphism*]{} between Lie groupoids is a pair $(F, f)$ of smooth maps $F : G_1 \rightarrow G_2$ and $f: M_1 \rightarrow M_2$ which commute with all the structure maps of $G_1$ and $G_2$. In other words, $$\alpha_2 \circ F = f \circ \alpha_1, ~ ~ \beta_2 \circ F = f \circ \beta_1, ~ ~ \text{and} ~ ~ F(g_1 h_1) = F(g_1)F(h_1)$$ for all $(g_1,h_1) \in (G_1)_{(2)}.$

Let $G \rightrightarrows M$ be a Lie groupoid. A [*Lie subgroupoid*]{} of it is a Lie groupoid $ H \rightrightarrows N$ together with injective immersions $i : H \rightarrow G$ and $i_0 : N \rightarrow M$ such that $(i, i_0)$ is a Lie groupoid morphism.

Let $G \rightrightarrows M$ be a Lie groupoid. A submanifold $\mathcal{K}$ of $G$ is called a [*bisection*]{} of the Lie groupoid, if $\alpha|_{\mathcal{K}} : \mathcal{K} \rightarrow M$ and $\beta|_{\mathcal{K}} : \mathcal{K} \rightarrow M$ are diffeomorphisms.

The existence of local bisections through any point $g \in G$ is always guaranted. The space of bisections $\mathcal{B}(G)$ form an infinite dimensional (Fréchet) Lie group under the multiplication of subsets induced from the partial multiplication of $G$. Note that the left (right) multiplication is defined only on $\alpha$-fibers ($\beta$-fibers), therefore, we can not define a diffeomorphism of $G$ using left (right) multiplication by an element, like a Lie group. However we can do so by using bisection instead of an element. Given a bisection $\mathcal{K} \in \mathcal{B}(G)$, let $l_{\mathcal{K}}$ and $r_{\mathcal{K}}$ be the diffeomorphisms on $G$ defined by $$l_{\mathcal{K}} (h) = gh, \hspace{0.15cm} \text{where} \hspace{0.15cm} g \in \mathcal{K} \hspace{0.15cm} \text{is the unique element such that} \hspace{0.15cm} \beta(g) = \alpha(h)$$ and $$r_{\mathcal{K}} (h) = hg', \hspace{0.15cm} \text{where} \hspace{0.15cm} g' \in \mathcal{K} \hspace{0.15cm} \text{is the unique element such that} \hspace{0.15cm} \alpha(g') = \beta(h).$$

Suppose $\mathcal{K}$ is any (local) bisection of $G$ through $g \in G$. Then the restriction of the map $l_{\mathcal{K}}$ to $\alpha^{-1} (\beta(g))$ is the left translation $l_g$ by $g$: $$l_g : \alpha^{-1}(\beta(g)) \rightarrow \alpha^{-1}(\alpha(g)), ~ h \mapsto gh .$$

Then we have the following result [@xu].

\[left-invariant\] Let $G \rightrightarrows M$ be a Lie groupoid and $P$ be an $n$-vector field on $G$. Suppose for any $g \in G$ with $\beta(g) = u$, $P$ satisfies $P(g) = (l_{\mathcal{G}})_* P(\epsilon_u)$, where $\mathcal{G} \in \mathcal{B}(G)$ is any arbitrary bisection through the point $g$. Then $P$ is left invariant.

A [*Lie algebroid*]{} $(A, [~, ~], a)$ over a smooth manifold M is a smooth vector bundle $A$ over M together with a Lie algebra structure $[~, ~]$ on the space $\Gamma{A}$ of the smooth sections of $A$ and a bundle map $a : A \rightarrow T M $ , called the [*anchor*]{}, such that

1.  the induced map $ a : \Gamma{A} \rightarrow \mathcal{X}^1(M) $ is a Lie algebra homomorphism, where $\mathcal{X}^1(M)$ is the usual Lie algebra of vector fields on $M$.

2.  For any $ X, Y \in \Gamma{A} $ and $f \in C^\infty (M)$, we have $$[X, f Y ] = f [X, Y ] + (a(X)f )Y.$$

We may denote a Lie algebroid simply by $A$, when all the structures are understood. Any Lie algebra is a Lie algebroid over a point with zero anchor. The tangent bundle of any smooth manifold is a Lie algebroid with usual Lie bracket of vector fields and identity as anchor.\
[**Lie algebroid of a Lie groupoid.**]{} Given a Lie groupoid $G \rightrightarrows M$, its Lie algebroid consists of the vector bundle $AG \rightarrow M$ whose fiber at $x \in M$ coincides with the tangent space at the unit element $\epsilon_x$ of the $\alpha$-fiber at $x$. Then the space of sections of $AG$ can be identified with the left invariant vector fields $$\begin{aligned}
 \mathcal{X}^1_{\text{inv}}(G) = \{ X \in \Gamma (T^\alpha G) = \Gamma (\text{ker} (d\alpha))| X_{gh} = (l_g)_* X_h, \forall (g, h) \in  G_{(2)} \}\end{aligned}$$ on $G$. Since the space of left invariant vector fields on $G$ is closed under the Lie bracket, therefore it defines a Lie bracket on $\Gamma AG$. The anchor $a$ of $AG$ is defined to be the differential of the target map $\beta$ restricted to $AG$.

Let $AG $ be the Lie algebroid of the Lie groupoid $G \rightrightarrows M$. Given any $X \in \Gamma AG$, let $\overleftarrow{X}$ be the corresponding left invariant vector field on $G$. Then there exists an $\epsilon > 0$ and a $1$-parameter family of transformations $\phi_t$ $(|t| < \epsilon)$, generated by $\overleftarrow{X}$ ([@mac-book]). Suppose each $\phi_t$ is defined on all of $M$, where $M$ is identified with a closed embedded submanifold of $G$ via the unit map. We denote the image of $M$ via $\phi_t$ by exp $tX$. Then exp $tX$ is a bisection of the groupoid (for all $|t| < \epsilon$) and satisfies $1$-parameter group like conditions, namely $$\begin{aligned}
 \text{exp} (t+s) X =  \text{exp} \hspace*{0.1cm} tX \cdot \text{exp} \hspace*{0.1cm}sX, \hspace*{1cm} \text {whenever} \hspace*{0.5cm} |t| , |s| , |t+s| < \epsilon,\end{aligned}$$ where on the right hand side, we used the multiplication of bisections.

Coisotropic submanifolds
========================

Let $M$ be a manifold and $\Pi \in \mathcal{X}^n(M)$ be a $n$-vector field on $M$. Let $$\begin{aligned}
 \Pi^{\sharp} : {\bigwedge}^{n-1}T^*M \rightarrow TM\end{aligned}$$ be the induced bundle map given by $$\begin{aligned}
 \langle \beta, \Pi^{\sharp} (\alpha_1 \wedge \cdots \wedge \alpha_{n-1}) \rangle = \Pi (\alpha_1, \ldots, \alpha_{n-1}, \beta)\end{aligned}$$ for all $\alpha_1, \ldots, \alpha_{n-1}, \beta \in T_x^* M,$ $x \in M$.

We recall the following definition from [@pont-geng-xu].

A submanifold $C \hookrightarrow M$ is said to be [*coisotropic*]{} with respect to $\Pi$, if $$\begin{aligned}
 \Pi^{\sharp} ({\bigwedge}^{n-1} (TC)^0) \subset TC\end{aligned}$$ where $$(TC)^0_x = \{ \alpha \in  T_x^* M | \hspace*{0.1cm} \alpha (v) = 0, \forall v \in T_xC\},~~x \in C,$$ or equivalently, $$\begin{aligned}
 \Pi_x (\alpha_1, \ldots, \alpha_n) = 0, \forall \alpha_i \in (TC)^0_x, x \in C.\end{aligned}$$

We have the following easy observation for coisotropic submanifolds of a Nambu-Poisson manifold.

Let $(M, \Pi)$ be a Nambu-Poisson manifold of order $n$ and $C$ be a closed embedded submanifold of $M$. Let $\mathcal{I} (C) = \{ f \in C^\infty(M)\big| f|_C \equiv 0 \}$ denote the vanishing ideal of $C$. Then the followings are equivalent:

1.  $C$ is a coisotropic submanifold;

2.  $\mathcal{I} (C)$ is a Nambu-Poisson subalgebra;

3.  for every $f_1, \ldots, f_{n-1} \in \mathcal{I} (C)$, the Hamiltonian vector field $X_{f_1...f_{n-1}}$ is tangent to $C$.

\(1) $\Rightarrow$ (2) Let $f_1, \ldots, f_n \in \mathcal{I} (C) $. Then for any $x \in C$, $d_xf_i \in (TC)^0_x$, for all $i=1, \ldots, n.$ Now since $C$ is a coisotropic submanifold, we have $$\begin{aligned}
 \{f_1, \ldots, f_n\}(x) = \Pi_x (d_xf_1, \ldots, d_xf_n) = 0,~ \forall x \in C.\end{aligned}$$ Hence $\{f_1, \ldots, f_n\} \in \mathcal{I} (C)$. Therefore $\mathcal{I} (C)$ is a Nambu-Poisson subalgebra.

\(2) $\Rightarrow$ (3) Let $f_1, \ldots, f_{n-1} \in \mathcal{I} (C)$ and $x \in C$. Let $\alpha \in (TC)^0_x.$ Then there exists a function $g$ vanishing on $C$ such that $d_xg = \alpha$. Since $\mathcal{I} (C)$ is a Nambu-Poisson subalgebra, we have $$\begin{aligned}
 \{f_1, \ldots, f_{n-1}, g \}(x) = 0.\end{aligned}$$ Thus, $$\begin{aligned}
 X_{f_1\ldots f_{n-1}}\big|_x (\alpha) = X_{f_1\ldots f_{n-1}}\big|_x (d_xg) = \{f_1, \ldots, f_{n-1}, g \}(x) = 0\end{aligned}$$ and consequently, $X_{f_1\ldots f_{n-1}}$ is tangent to $C$.

\(3) $\Rightarrow$ (1) Let $x \in C$ and $\alpha_1, \ldots, \alpha_n \in (TC)^0_x.$ Then there exist functions $f_1, \ldots, f_n \in \mathcal{I}(C)$ such that $d_xf_i = \alpha_i,$ $\forall i = 1, \ldots, n.$ Therefore, $$\begin{aligned}
 \Pi_x (\alpha_1, \ldots, \alpha_n) = \Pi_x (d_xf_1, \ldots, d_xf_n) = X_{f_1...f_{n-1}} \big|_x (d_xf_n) = 0.\end{aligned}$$ Hence $C$ is a coisotropic submanifold of $M$.

Let $(M, \Pi_M)$ and $(N, \Pi_N)$ be two Nambu-Poisson manifolds of same order $n$ and $C \hookrightarrow N$ be a coisotropic submanifold of $N$ with respect to $\Pi_N$. If $\phi: M \rightarrow N$ is a Nambu-Poisson map transverse to $C$, then $\phi^{-1}(C)$ is a coisotropic submanifold of $M$ with respect to $\Pi_M$ (the result holds true for manifolds with $n$-vector fields such that $\phi_* \Pi_M = \Pi_N$).

Since $\phi$ is transverse to $C$, therefore $\phi^{-1}(C)$ is a submanifold of $M$. Moreover $$\begin{aligned}
 T(\phi^{-1}(C)) = (T \phi)^{-1} TC.\end{aligned}$$ Therefore $T(\phi^{-1}(C))^0 = (T \phi)^{*} (TC)^0$. Observe that $$\begin{aligned}
 T\phi (\Pi_M^{\sharp} \big( {\bigwedge}^{n-1} T(\phi^{-1}(C))^0  \big)) =& T\phi ( \Pi_M^{\sharp} \big((T \phi)^{*} {\bigwedge}^{n-1} (TC)^0 \big))\\
=& \Pi_N^{\sharp} ({\bigwedge}^{n-1} (TC)^0)\\
\subseteq & TC.\end{aligned}$$ Thus, $$\begin{aligned}
 \Pi_M^{\sharp} \big( {\bigwedge}^{n-1} T(\phi^{-1}(C))^0  \big) \subseteq (T\phi)^{-1} TC = T(\phi^{-1}(C))\end{aligned}$$ and hence $\phi^{-1}(C)$ is coisotropic with respect to $\Pi_M.$

\[image-coiso\] Let $\phi: (M, \Pi_M) \rightarrow (N, \Pi_N)$ be a Nambu Poisson map between two Nambu-Poisson manifolds $(M, \Pi_M)$ and $(N, \Pi_N)$ and $C\hookrightarrow M$ be a coisotropic submanifold of $M$. Assume that $\phi(C)$ is a submanifold of $N$. Then $\phi(C)$ is a coisotropic submanifold of $N$ (the result holds true for manifolds with $n$-vector fields such that $\phi_{*}\Pi_M = \Pi_N$).

We have $T(\phi(C)) \supseteq T\phi (TC)$ and $(T\phi)^* (T(\phi(C)))^0 \subseteq (TC)^0$. Therefore, $$\begin{aligned}
 \Pi_N^{\sharp} ({\bigwedge}^{n-1} T(\phi(C))^0) =& T\phi (\Pi_M^{\sharp} ((T\phi)^* {\bigwedge}^{n-1} T(\phi(C))^0)) \hspace*{1cm}(\text{since ~} \phi \text{~ is a N-P map})\\
\subseteq & T\phi (\Pi_M^{\sharp} ({\bigwedge}^{n-1} (TC)^0))\\
\subseteq & T\phi (TC) \hspace*{1cm}(\text{since~} C \hookrightarrow M \text{~ is coisotropic})\\
\subseteq & T(\phi(C))\end{aligned}$$ which shows that $\phi(C)$ is a coisotropic submanifold of $N$.

Using the terminology of coisotropic submanifold with respect to any multivector field allows us to extend the results of Weinstein [@wein] from Poisson bivector field to Nambu-Poisson tensor or more generally to any multivector field.

\[nambu map-coiso\] Let $(M, \Pi_M)$ and $(N, \Pi_N)$ be two manifolds with $n$-vector fields and $\phi: M \rightarrow N$ be a smooth map. Then $\phi$ is a $(\Pi_M, \Pi_N)$-map, that is $\phi_* \Pi_M = \Pi_N$ if and only if its graph $$\begin{aligned}
 \text{Gr}(\phi) = \{(m, \phi(m))| m \in M \}\end{aligned}$$ is a coisotropic submanifold of $M \times N$ with respect to $\Pi_M \oplus (-1)^{n-1} {\Pi}_N$.

Let $C = \text{Gr}(\phi) \subset M \times N$. Then $C$ is a closed embedded submanifold of $M \times N.$ Note that, a tangent vector to the graph consist of a pair $(v_m, (T\phi)(v_m))$, where $m \in M$, $v_m \in T_mM$. Therefore, $(TC)^0$ consists of a pair of covectors $(-(T\phi)^*\psi, \psi)$, where $\psi \in T_{\phi(m)}^*N.$ Therefore, Gr($\phi)$ is a coisotropic submanifold of $M \times N$ with respect to $\Pi_M \oplus (-1)^{n-1} {\Pi}_N$ if and only if $(\Pi_M^{\sharp} \times (-1)^{n-1} \Pi_N^{\sharp})$ maps $(-(T\phi)^*\psi_1, \psi_1) \wedge \cdots \wedge (-(T\phi)^*\psi_{n-1}, \psi_{n-1}) $ into $TC$, for all $\psi_1, \ldots, \psi_{n-1} \in T^*_{\phi(m)}N$ and $m \in M$. In other words, $$\begin{aligned}
(T\phi) \bigg(\Pi_M^{\sharp} (- (T\phi)^*\psi_1, \ldots, - (T\phi)^*\psi_{n-1})\bigg) = (-1)^{n-1} {\Pi}_N^{\sharp} (\psi_1, \ldots, \psi_{n-1})\end{aligned}$$ that is, $$\begin{aligned}
(T\phi) \bigg(\Pi_M^{\sharp} ( (T\phi)^*\psi_1, \ldots,  (T\phi)^*\psi_{n-1})\bigg) =  {\Pi}_N^{\sharp} (\psi_1, \ldots, \psi_{n-1}).\end{aligned}$$ This is equivalent to the condition that $\phi$ is a $(\Pi_M, \Pi_N)$-map.

Let $(M, \Pi_M)$ be a Nambu-Poisson manifold of order $n$ and $\phi: M \rightarrow N$ be a smooth surjective map. If there exist a Nambu-Poisson structure $\Pi_N$ (of order $n$) on $N$ which makes $\phi$ into a Nambu-Poisson map, then $\Pi_N$ is called the Nambu-Poisson structure [*coinduced*]{} by the mapping $\phi$.

The following is a characterization of coinduced Nambu-Poisson structure.

\[coinduced\] Let $(M, \Pi_M)$ be a Nambu-Poisson manifold of order $n$ and $\phi: M \rightarrow N$ be a smooth surjective map from $M$ to some manifold $N$. Then $N$ has a Nambu-Poisson structure coinduced by $\phi$ if and only if for all $f_1, \ldots, f_{n} \in C^\infty(N)$, the function $\{\phi^*f_1, \ldots, \phi^*f_n\}_M$ is constant along the fibers of $\phi$.

Let $f_1, \ldots, f_n \in C^\infty(N)$. If the function $\{ \phi^*f_1, \ldots, \phi^*f_n \}_M$ is constant along the $\phi$-fibers, then there exists a function on $N$, which we denote by $\{f_1, \ldots, f_n\}_N$ such that $\{\phi^*f_1, \ldots, \phi^*f_n\}_M = \phi^* \{f_1, \ldots, f_n\}_N $. Clearly this bracket defines a coinduced Nambu-Poisson structure on $N$.

Conversely, suppose that there is a Nambu-Poisson bracket $\{~, \ldots, ~\}_N$ on $N$ coinduced by $\phi$. Then for any $y \in N,$ $$\begin{aligned}
\{\phi^*f_1, \ldots, \phi^*f_n\}_M(\phi^{-1}\{y\}) =& (\phi^* \{f_1, \ldots, f_n\}_N )(\phi^{-1}y)\\
=& \{f_1, \ldots, f_n\}_N (y)\end{aligned}$$ proving $\{\phi^*f_1, \ldots, \phi^*f_n\}_M$ is constant along the $\phi$-fibers.

\[rem-coinduced\] Let $(M, \Pi_M)$ be a manifold with an $n$-vector field and $\phi: M \rightarrow N$ be a smooth map. Then there exists an $n$-vector field $\Pi_N$ on $N$ such that $\phi$ is a $(\Pi_M, \Pi_N)$-map if and only if for all $f_1, \ldots, f_n \in C^\infty(N)$, the function $\{\phi^*f_1, \ldots, \phi^*f_n\}_M$ is constant along the fibers of $\phi$.

Let $(M, \Pi_M)$ be a Nambu-Poisson manifold and $\phi : M \rightarrow N$ be a surjective submersion with connected fibers. Let $\text{ker} \hspace*{0.1cm} \phi_*(m)$ is spanned by local Hamiltonian vector fields (that is, $\text{ker} \hspace*{0.1cm} \phi_*(m) \subset \mathcal{D}_mM $), for all $m \in M$. Then $N$ has a Nambu-Poisson structure coinduced by $\phi.$

Since $\phi$ is a submersion, the fibers of $\phi$ are submanifolds of $M$. Then for $y \in N,$ $\phi^{-1}(\{y\}) = C$ is a submanifold of $M$. Let $g_1, \ldots, g_{n-1}$ be locally defined functions on $M$ such that $X_{g_1...g_{n-1}} \in \text{ker} \hspace*{0.1cm} \phi_{*}$. Let $f_1, \ldots, f_n \in C^\infty(N).$ To prove that $\{\phi^*f_1, \ldots, \phi^*f_n\}$ is constant on the fibers, it is enough to prove that $$\begin{aligned}
 X_{g_1...g_{n-1}} \{\phi^*f_1, \ldots, \phi^*f_n\} = 0.\end{aligned}$$ Note that $$\begin{aligned}
 X_{g_1...g_{n-1}} \{\phi^*f_1, \ldots, \phi^*f_n\} = \sum_{k=1}^n \{\phi^*f_1, \ldots, X_{g_1...g_{n-1}} (\phi^*f_k), \ldots, \phi^*f_n\}\end{aligned}$$ and the functions $\phi^*f_i$ are constant along the fibers. Hence by the Proposition \[coinduced\], there exists a coinduced Nambu-Poisson structure on $N.$

\[coinduced-coiso\] Let $(M, \Pi_M)$ be a manifold with an $n$-vector field and $\phi: M \rightarrow N$ be a surjective submersion. Then $N$ has an (unique) $n$-vector field $\Pi_N$ such that $\phi$ is a $(\Pi_M, \Pi_N)$-map if and only if $R(\phi) = \{(x,y) \in M \times M|\phi(x)=\phi(y)\}$ is a coisotropic submanifold of $M \times M$ with respect to $\Pi_M \oplus (-1)^{n-1}{\Pi}_M$.

Note that $R(\phi) = (\phi \times \phi)^{-1} (\Delta_N)$, where $\Delta_N$ is the diagonal of $N \times N$. Since $\phi$ is surjective submersion $R(\phi)$ is a submanifold of $M \times M$. Moreover, for $(x,y) \in R(\phi)$ $$\begin{aligned}
 T_{(x,y)} (R(\phi)) = \{(X, Y)\in T_xM \times T_yM | (T\phi)_x (X) = (T\phi)_y (Y)\}.\end{aligned}$$ Therefore, $T(R(\phi))^0$ consists of covectors $(-(T\phi)_x^*\psi, (T\phi)_y^*\psi)$, where $\psi \in T_{\phi(x)}^*N$.

Thus, $R(\phi)$ be a coisotropic submanifold of $M \times M$ with respect to $\Pi_M \oplus (-1)^{n-1}{\Pi}_M$ if and only if for all $\psi_1, \ldots, \psi_{n-1} \in T_{\phi(x)}^*N$ and $(x, y) \in R(\phi),$ $\Pi_M^{\sharp} \oplus (-1)^{n-1}{\Pi}_M^{\sharp}$ maps $$(-(T\phi)_x^*\psi_1, (T\phi)_y^*\psi_1) \wedge \cdots \wedge (-(T\phi)_x^*\psi_{n-1}, (T\phi)_y^*\psi_{n-1})$$ into $T(R(\phi)).$ That is $$(T\phi)_x \Pi_M^{\sharp} (-(T\phi)_x^*\psi_1, \ldots, -(T\phi)_x^*\psi_{n-1} )= (-1)^{n-1} (T\phi)_y \Pi_M^{\sharp} ((T\phi)_y^*\psi_1, \ldots, (T\phi)_y^*\psi_{n-1}),$$ or equivalently, $$\begin{aligned}
 \label{firsteqn}
 (T\phi)_x \Pi_M^{\sharp} ((T\phi)_x^*\psi_1, \ldots, (T\phi)_x^*\psi_{n-1} ) =  (T\phi)_y \Pi_M^{\sharp} ((T\phi)_y^*\psi_1, \ldots, (T\phi)_y^*\psi_{n-1})\end{aligned}$$ holds. Let $f_1, \ldots ,f_n \in C^\infty(N)$ and $x \in M.$ Then $$\begin{aligned}
\{\phi^*f_1, \ldots , \phi^*f_n\}_M (x) =& \langle \Pi_M^{\sharp} (d_x (\phi^*f_1) \wedge \cdots \wedge d_x (\phi^*f_{n-1})), d_x (\phi^*f_n) \rangle\\
 =& \langle \Pi_M^{\sharp} \big((T\phi)_x^* \psi_1 \wedge \cdots \wedge (T\phi)_x^* \psi_{n-1}\big), (T\phi)_x^* \psi_n \rangle\\
 =& \langle (T\phi)_x \Pi_M^{\sharp} \big((T\phi)_x^* \psi_1 \wedge \cdots \wedge (T\phi)_x^* \psi_{n-1}\big), \psi_n \rangle\end{aligned}$$ where $\psi_i = d_{\phi(x)}f_i = d_{\phi(y)}f_i\in T_{\phi(x)}^*N$, for all $1 \leqslant i \leqslant n.$ It follows from the Equation (\[firsteqn\]) that the function $\{\phi^*f_1, \ldots , \phi^*f_n\}_M$ is constant along the $\phi$-fibers if and only if $R(\phi)$ is a coisotropic submanifold of $M \times M$ with respect to $\Pi_M^{\sharp} \oplus (-1)^{n-1}{\Pi}_M^{\sharp}$. Hence the result follows by the Remark \[rem-coinduced\]. The uniqueness follows from the surjectivity of $\phi.$

Nambu-Lie groupoids
===================

In this section, we recall the definition of multiplicative multivector fields on Lie groupoid ([@pont-geng-xu]) and define Nambu-Lie groupoid (of order $n$) as a Lie groupoid with a multiplicative $n$-vector field which is also a Nambu-Poisson tensor.

Let $G \rightrightarrows M$ be a Lie groupoid and $\Pi \in \mathcal{X}^n(G)$ be an $n$-vector field on $G$. Then $\Pi$ is called [*multiplicative*]{} if the graph of the groupoid multiplication $$\begin{aligned}
 \{ (g, h, gh) \in G \times G \times G | ~ \beta(g) = \alpha(h)\}\end{aligned}$$ is a coisotropic submanifold of $G \times G \times G$ with respect to $\Pi \oplus \Pi \oplus (-1)^{n-1} \Pi$.

Then we have the following characterization of multiplicative multivector fields [@pont-geng-xu]:

\[multiplicative\] Let $G \rightrightarrows M$ be a Lie groupoid and $\Pi \in \mathcal{X}^n (G)$ be an $n$-vector field on $G$. Then $\Pi$ is multiplicative if and only if the following conditions are satisfied.

1.  $\Pi$ is an affine tensor. In other words $$\begin{aligned}
     \Pi(gh) = (r_{\mathcal{H}})_* \Pi(g) + (l_{\mathcal{G}})_* \Pi(h)  - (r_{\mathcal{H}})_* (l_{\mathcal{G}})_* \Pi(u)\end{aligned}$$ where $u = \beta(g) = \alpha(h)$ and $\mathcal{G}, \mathcal{H}$ are (local) bisections through the points $g, h$ respectively.

2.  $M$ is a coisotropic submanifold of $G$ with respect to $\Pi$.

3.  For all $g \in G$, $\alpha_* \Pi(g)$ and $\beta_* \Pi(g)$ depend only on the base points $\alpha(g)$ and $\beta(g)$ respectively.

4.  For all $f, f' \in C^\infty(M)$, the $(n-2)$-vector field $\iota_{d(\alpha^*f) \wedge d(\beta^*f')} \Pi$ is zero. In other words, $$\begin{aligned}
     \{~, \ldots, \alpha^*f, \beta^*f' \} = 0.\end{aligned}$$

5.  For all $f_1, \ldots, f_k \in C^\infty(M)$, $\iota_{d(\beta^*f_1) \wedge \cdots \wedge d(\beta^*f_k)} \Pi$ is a left invariant $(n-k)$-vector field on $G$, $1 \leqslant k < n.$

\[lie-gp-multiplicative\] Suppose $G$ be a Lie group considered as a Lie groupoid over a point. Then the conditions [*(3)*]{} - [*(5)*]{} of the Theorem \[multiplicative\] are satisfied automatically. The condition [*(2)*]{} implies that $\Pi (e) = 0$ (where $e$ is the identity element of the group), which together with condition [*(1)*]{} implies that $\Pi$ satisfies the usual multiplicativity condition $$\Pi (gh) = (r_h)_* \Pi(g) + (l_g)_* \Pi(h) .$$

\[nambu-lie groupoid\] A [*Nambu-Lie groupoid of order $n$*]{} is a Lie groupoid $G \rightrightarrows M$ with a multiplicative Nambu tensor $\Pi \in \mathcal{X}^n(G)$ of order $n.$

A Nambu Lie groupoid (of order $n$) will be denoted by $(G \rightrightarrows M, \Pi) $.

\[exam-nlg\]

1.  Poisson groupoids [@wein] are examples of Nambu-Lie groupoids with $n=2.$

2.  Any Lie groupoid with zero Nambu structure is a Nambu-Lie groupoid.

3.  Let $(G, \Pi)$ be a Nambu-Lie group (of order $n$) [@vais]. Thus $G$ is a Lie group equipped with a Nambu structure $\Pi$ of order $n$ on $G$ such that $$\begin{aligned}
     \Pi(gh) = (r_h)_* \Pi(g) + (l_g)_* \Pi(h)\end{aligned}$$ for all $g, h \in G$. Note that the right hand side of the above equality is equal to $m_* (\Pi(g), \Pi(h)),$ where $m_* : \bigwedge^n T_{(g,h)} (G \times G) \rightarrow \bigwedge^n T_{gh}G$ is the map induced by the multiplication map $m : G \times G \rightarrow G$. Therefore, $$\begin{aligned}
     \Pi(gh) = m_* (\Pi(g), \Pi(h)).\end{aligned}$$ Thus, the group multiplication map $m : G \times G \rightarrow G$ is a $(\Pi \oplus \Pi, \Pi)$-map. Therefore, by the Proposition \[nambu map-coiso\], the graph of the group multiplication map is a coisotropic submanifold of $G \times G \times G$ with respect to $\Pi \oplus \Pi \oplus (-1)^{n-1} \Pi$. Hence $(G, \Pi)$ is a Nambu-Lie groupoid over a point. Conversely, if $(G, \Pi)$ is a Nambu-Lie groupoid over a point, then the group multiplication map $m : G \times G \rightarrow G$ is a $(\Pi \oplus \Pi, \Pi)$-map. Hence $(G, \Pi)$ is a Nambu-Lie group in the sense of [@vais]. One can also see the equivalence between Nambu-Lie groupoid over a point and Nambu-Lie group by using Remark \[lie-gp-multiplicative\].

For a Poisson groupoid the following facts are well known [@wein].

-   The groupoid inversion map is a anti-Poisson map.

-   The Poisson structure on the total space induces a Poisson structure on the base such that the source map is a Poisson map and the target map is a anti-Poisson map.

In the next proposition we generalize the above facts to the Nambu-Poisson setting.

\[inverse-basenambu\] Let $(G \rightrightarrows M , \Pi)$ be a Nambu-Lie groupoid. Then

1.  The inverse map $i : G \rightarrow G$, $g \mapsto g^{-1}$ is an anti-Nambu Poisson map.

2.  There is a unique Nambu-Poisson structure on $M$ which we denote by $\Pi_M$ for which $\alpha$ is a Nambu-Poisson map and $\beta$ is an anti Nambu-Poisson map.

\(1) It is proved in [@pont-geng-xu] that given a Lie groupoid $G \rightrightarrows M$ with multiplicative $n$-vector field $\Pi \in \mathcal{X}^n(G)$, the groupoid inversion map $i : G \rightarrow G$ satisfies $$\begin{aligned}
 i_* \Pi = (-1)^{n-1} \Pi.\end{aligned}$$ Hence the result follows as $\Pi$ is a Nambu tensor.

\(2) Let $f_1, \ldots, f_n \in C^\infty(M)$ be any functions on $M$. Then for any $g \in G$, we have $$\begin{aligned}
\{\alpha^* f_1, \ldots, \alpha^* f_n \} (g) =& \Pi (g) (d_g (\alpha^* f_1), \ldots, d_g (\alpha^* f_n))\\
 =& \Pi(g) (\alpha^* (d_{\alpha(g)} f_1), \ldots, \alpha^* (d_{\alpha(g)} f_n))\\
 =& \alpha_* \Pi (g) (d_{\alpha(g)} f_1, \ldots, d_{\alpha(g)} f_n).\end{aligned}$$ Since $\alpha_* \Pi(g)$ depends only on the value of $\alpha(g)$, it follows that the function $\{ \alpha^* f_1, \ldots, \alpha^* f_n \}$ is constant on the $\alpha$-fibers. Therefore, by the Proposition \[coinduced\], there exists a Nambu-Poisson structure $\Pi_M$ with the induced bracket denoted by $\{ ~, \ldots, ~\}_M,$ on $M$ for which $\alpha$ is a Nambu-Poisson map. Since $\beta = \alpha \circ i$ and $i$ is anti Nambu-Poisson, therefore $\beta$ is an anti Nambu-Poisson map.

Consider the map $(\alpha,\beta): G \rightarrow M \times M.$ Since we have $\alpha_{*} \Pi = \Pi_M$ and $\beta_{*}\Pi = (-1)^{n-1} \Pi_M$, using property [*(4)*]{} of the Theorem \[multiplicative\] we obtain $$\begin{aligned}
(\alpha,\beta)_{*} \Pi = \alpha_* \Pi \oplus \beta_* \Pi = \Pi_M \oplus (-1)^{n-1} \Pi_M.\end{aligned}$$

Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid. If the orbit space $M/\sim$ is a smooth manifold, then $M/\sim$ carries a Nambu-Poisson structure such that the projection $q: M \rightarrow M/\sim $ is a Nambu-Poisson map.

Let $\Pi_M$ be the induced Nambu structure on the base $M$. For the projection map $q: M \rightarrow M/\sim $, we have $$\begin{aligned}
 R(q) =& \{(x,y) \in M \times M| q(x) = q(y)\}\\
=& \{(\alpha(g), \beta(g))| g \in G\}\\
=& (\alpha,\beta)(G).\end{aligned}$$ Consider $G$ as a coisotropic submanifold of $G$ with respect to $\Pi$ and also consider the map $(\alpha, \beta) : G \rightarrow M \times M$. By the above Remark we have $(\alpha, \beta)_* \Pi = \Pi_M \oplus (-1)^{n-1} \Pi_M.$ Therefore, by the Proposition \[image-coiso\], $R(q) = (\alpha,\beta)(G)$ is a coisotropic submanifold of $M \times M$ with respect to $\Pi_M \oplus (-1)^{n-1}\Pi_M.$ Hence the result follows from the Proposition \[coinduced-coiso\].

Infinitesimal form of Nambu-Lie groupoid
========================================

The aim of this section, is to study the infinitesimal form of a Nambu-Lie groupoid. We show that if $(G \rightrightarrows M, \Pi)$ is a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$, then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ introduced in [@bas-bas-das-muk]. Before proceeding further, let us briefly recall from [@bas-bas-das-muk] the notion of a weak Lie-Filippov bialgebroid.

Lie bialgebroids are generalization of both Poisson manifolds and Lie bialgebras. Recall that a Lie bialgebroid, introduced by Mackenzie and Xu [@mac-xu] is also the infinitesimal form of a Poisson groupoid. It is defined as a pair $(A, A^\ast)$ of Lie algebroids in duality, where the Lie bracket of $A$ satisfies the following compatibility condition expressed in terms of the differential $d_\ast$ on $\Gamma (\bigwedge^\bullet A)$ $$d_\ast[X, Y] = [d_\ast X, Y] + [X , d_\ast Y],$$ for all $X,~ Y \in \Gamma A.$

We note that if $M$ is a Poisson manifold then the Lie algebroid structures on $TM$ and $T^\ast M$ form a Lie bialgebroid. On the other hand, it is well known [@kosmann; @mac-xu] that if $(A, A^\ast)$ is a Lie bialgebroid over a smooth manifold $M$ then there is a canonical Poisson structure on the base manifold $M$.

Thus it is natural to ask the following question which was posed in [@bas-bas-das-muk]:\
[*Does there exist some notion of bialgebroid associated to a Nambu-Poisson manifold of order $n > 2$* ]{}?

To answer this question, the authors [@bas-bas-das-muk] introduced the notion of weak Lie-Filippov bialgebroid.

It is well known [@gra-mar; @vais] that for a Nambu-Poisson manifold $M$ of order $n \geqslant 2$, the space $\Omega^1(M)$ of $1$-forms admits an $n$-ary bracket, called [*Nambu-form bracket*]{}, such that the bracket satisfies almost all the properties of an $n$-Lie algebra (also known as Filippov algebra of order $n$) bracket [@fil] except that the Fundamental identity is satisfied only in a restricted sense as described below.

Let $(M, \{~, \ldots, ~\})$ be a Nambu-Poisson manifold of order $n$ with associated Nambu-Poisson tensor $\Pi$. Then one can define the Nambu form-bracket on the space of $1$-forms $$\begin{aligned}
 [~, \ldots, ~]: \Omega^1(M) \times  \cdots \times \Omega^1(M) \rightarrow \Omega^1(M)\end{aligned}$$ by the following $$\begin{aligned}
 [\alpha_1, \ldots, \alpha_n] = \sum_{k=1}^n (-1)^{n-k} \mathcal{L}_{\Pi^{\sharp}(\alpha_1 \wedge\cdots \wedge \hat{\alpha}_k \wedge\cdots  \wedge \alpha_n)} \alpha_k - (n-1) d (\Pi(\alpha_1, \ldots, \alpha_n))\end{aligned}$$ $$\begin{aligned}
\label{nambu-bracket}
\hspace*{1cm} = d (\Pi(\alpha_1, \ldots, \alpha_n)) + \sum_{k=1}^n (-1)^{n-k} \iota_{\Pi^{\sharp}(\alpha_1 \wedge\cdots \wedge \hat{\alpha}_k \wedge\cdots \wedge \alpha_n)} d\alpha_k\end{aligned}$$ for $\alpha_i \in \Omega^1(M), i=1,\ldots ,n.$ Here $\hat{\alpha}_k$ in a monomial $\alpha_1 \wedge \cdots \wedge \hat{\alpha}_k \wedge \cdots \wedge \alpha_n$ means that the symbol $\alpha_k$ is missing in the monomial. The above bracket satisfies the following properties ([@vais]).

1.  The bracket is skew-symmetric.

2.  $[df_1, \ldots, df_n] = d \{f_1, \ldots, f_n \}$.

3.  $[\alpha_1,\ldots, \alpha_{n-1}, f \alpha_n] = f [\alpha_1,\ldots, \alpha_{n-1},  \alpha_n] + \Pi ^{\sharp} (\alpha_1 \wedge\cdots \wedge \alpha_{n-1})(f) \alpha_n$.

4.  The bracket satisfies the Fundamental identity $$\begin{aligned}
     [\alpha_1, \ldots, \alpha_{n-1}, [\beta_1, \ldots, \beta_n]] = \sum_{k=1}^n [\beta_1, \ldots , \beta_{k-1},[\alpha_1, \ldots, \alpha_{n-1}, \beta_k], \ldots, \beta_n]\end{aligned}$$ whenever the $1$-forms $\alpha_i \in \Omega^1(M)$ are closed, $1 \leqslant i \leqslant n-1$ and for any $\beta_j$.

5.  $[\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \alpha_{n-1}), \Pi^{\sharp}(\beta_1 \wedge \cdots \wedge \beta_{n-1})]\\ 
    = \sum_{k=1}^{n-1} \Pi^{\sharp} (\beta_1 \wedge\cdots \wedge [\alpha_1,\ldots , \alpha_{n-1}, \beta_k] \wedge \cdots \wedge \beta_{n-1})$\
    for closed $1$-forms $\alpha_i \in \Omega^1 (M)$ and for any $1$-forms $\beta_j$.

The Nambu-form bracket on $\Omega^1(M)$, together with the usual Lie algebroid structure on $TM$ yields an example of a notion called a [*weak Lie-Filippov algebroid pair of order $n$*]{}, $n>2,$ on a smooth vector bundle (cf. Definition $5.5$, [@bas-bas-das-muk]).

In order to classify such structures, the authors formulate a notion of [*Nambu-Gerstenhaber algebra of order $n$*]{}. It turns out, weak-Lie-Filippov algebroid pair structures of order $n$, $n>2$, on a smooth vector bundle $A$ over $M$, are in bijective correspondence with Nambu-Gerstenhaber brackets of order $n$ on the graded commutative, associative algebra $\Gamma \bigwedge^\bullet A^*$ of multisections of $A^*$, where $A^*$ is the dual bundle (cf. Definition $5.7$, Theorem $5.8$, [@bas-bas-das-muk]).

Moreover, for a Nambu-Poisson manifold $M$ of order $n > 2$, the Nambu-Gerstenhaber bracket on $\Omega^\bullet(M)$, extending the Nambu-form bracket on $\Omega^1(M)$ satisfies certain suitable compatibility condition similar to the compatibility condition of a Lie bialgebroid. This motivates the authors to introduce the notion of a [*weak Lie-Filippov bialgebroid structure*]{} of order $n$ on a smooth vector bundle.

\[lie-fill-defn\] A [*weak Lie-Filippov bialgebroid of order $n>2$*]{} over a smooth manifold $M$ consists of a pair $(A, A^*)$, where $A$ is a smooth vector bundle over $M$ with dual bundle $A^*$ satisfying the following properties:

1.  $A$ is a Lie algebroid with $d_A$ being the differential of the Lie algebroid cohomology of $A$ with trivial representation;

2.  the space of smooth sections $\Gamma A^*$ admits a skew-symmetric $n$-ary bracket $$[~, \ldots ,~]: {\Gamma A^* \times \cdots \times \Gamma A^*} \longrightarrow \Gamma A^*$$ satisfying $$[\alpha_1, \ldots , \alpha_{n-1}, [\beta_1, \ldots , \beta_n]] = \sum_{k=1}^n [\beta_1, \ldots , \beta_{k-1}, [\alpha_1, \ldots , \alpha_{n-1}, \beta_k], \ldots ,\beta_n]$$ for all $d_A$-closed sections $\alpha_i \in \Gamma A^*,~ 1 \leqslant i \leqslant n-1$ and for any sections $\beta_j \in \Gamma A^*,~ 1\leqslant j\leqslant n;$

3.  there exists a vector bundle map $\rho : \bigwedge^{n-1}A^* \longrightarrow TM$, called the [*anchor*]{} of the pair $(A, A^*)$, such that the identity $$[\rho (\alpha_1 \wedge \cdots \wedge \alpha_{n-1}), \rho (\beta_1 \wedge \cdots \wedge \beta_{n-1})] = \sum_{k=1}^{n-1}\rho (\beta_1 \wedge \cdots \wedge [\alpha_1, \ldots , \alpha_{n-1}, \beta_k] \wedge \cdots \wedge \beta_{n-1})$$ holds for all $d_A$-closed sections $\alpha_i \in \Gamma A^*,~ 1\leqslant i \leqslant n-1$ and for any sections $\beta_j \in \Gamma A^*,~ 1\leqslant j\leqslant n-1;$

4.  for all sections $\alpha_i \in \Gamma A^*,~ 1\leqslant i \leqslant n$ and any $f \in C^\infty(M)$, $$[\alpha_1, \ldots , \alpha_{n-1}, f\alpha_n] = f [\alpha_1, \ldots , \alpha_{n-1}, \alpha_n] + \rho (\alpha_1 \wedge \cdots \wedge \alpha_{n-1})(f)\alpha_n$$ holds;

5.  the following compatibility condition holds: $$d_A[\alpha_1, \ldots , \alpha_n] = \sum_{k=1}^n [\alpha_1, \ldots , d_A\alpha_k, \ldots , \alpha_n],$$ for any $\alpha_i \in \Gamma A^\ast,$ $1 \leqslant i \leqslant n$, where the bracket $[~, \ldots , ~]$ on the right hand side is the graded extension of the bracket on $\Gamma A^\ast$.

A weak Lie-Filippov bialgebroid (of order $n$) over $M$ is denoted by $(A, A^*)$ when all the structures are understood. A Lie bialgebroid is a Lie-Filippov bialgebroid of order $2$ such that the conditions (2) and (3) of the above definition has no restriction on $\alpha$.

In [@bas-bas-das-muk], the authors have shown that for a Nambu-Poisson manifold $M$ of order $n > 2$, the pair $(TM, T^*M)$ is a weak Lie-Filippov bialgebroid of order $n$ (cf. Corollary $6.3$, [@bas-bas-das-muk]). It is also proved that if $(G, \Pi)$ is a Nambu-Lie group [@vais] of order $n$ with its Lie algebra $\mathfrak{g},$ then $(\mathfrak{g}, \mathfrak{g}^*)$ forms a (weak) Lie-Filippov bialgebroid of order $n$ over a Point.

It is known that the base of a Lie bialgebroid carries a natural Poisson structure. In [@bas-bas-das-muk] it has been extended to the Nambu-Poisson set up.

\[wlfb-base-nambu\]([@bas-bas-das-muk]) Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid (of order n) over $M$. Then the bracket $$\begin{aligned}
 \{f_1, \ldots, f_n\}_{(A, A^*)} := \rho (d_A f_1 \wedge \cdots \wedge d_A f_{n-1}) f_n\end{aligned}$$ defines a Nambu-Poisson structure of order n on $M$.

It is known that, given a coisotropic submanifold $C$ of a Poisson manifold $M$, the conormal bundle $(TC)^0 \rightarrow C$ is a Lie subalgebroid of the cotangent Lie algebroid $T^*M$ [@wein]. If $M$ is a Nambu-Poisson manifold of order $n$ ($n \geqslant 3$), the cotangent bundle $T^*M$ is not a Filippov algebroid. However we have the following useful result.

\[coiso-n-bracket\] Let $C$ be a closed embedded coisotropic submanifold of a Nambu-Poisson manifold $(M, \Pi)$ of order $n$. Then

1.  the bundle map $\Pi^{\sharp} : \bigwedge^{n-1}T^*M \rightarrow TM$ maps $\bigwedge^{n-1} (TC)^0$ to $TC$;

2.  the Nambu-form bracket on the space of $1$-forms $\Omega^1(M)$ can be restricted to the sections of the conormal bundle $ (TC)^0 \rightarrow C$.

The assertion $(1)$ follows from the definition of coisotropic submanifold. To prove $(2)$, let $\alpha_1, \ldots ,\alpha_n \in \Gamma (TC)^0.$ We extend them to $1$-forms on $M$, which we denote by the same notation. Let $X \in \mathcal{X}^1(M)$ be such that $X\big|_C$ is tangent to $C$. From the definition of Nambu-form bracket on $1$-forms, we have $$\begin{aligned}
\langle [\alpha_1, \ldots, \alpha_n], X \rangle = \sum_{k=1}^n (-1)^{n-k} \langle \mathcal{L}_{\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \widehat{\alpha}_k \wedge \cdots \wedge \alpha_n)} \alpha_k, X \rangle
- (n-1) \langle d (\Pi(\alpha_1,\ldots, \alpha_n)), X \rangle. \end{aligned}$$ Observe that $$\begin{aligned}
 \langle \mathcal{L}_{\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \widehat{\alpha}_k \wedge \cdots \wedge  \alpha_n)}  \alpha_k, X \rangle 
=& \mathcal{L}_{ \Pi^\sharp (\alpha_1 \wedge \cdots \wedge \widehat{\alpha}_k \wedge \cdots \wedge \alpha_n )} \langle \alpha_k, X \rangle \\
&- \langle \alpha_k, [\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \widehat{\alpha}_k \wedge \cdots \wedge \alpha_n), X]\rangle .\end{aligned}$$ This is zero on $C$, because,

-   $\langle \alpha_k, X\rangle$ is zero on $C$;

-   $\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \widehat{\alpha_k} \wedge \cdots \wedge \alpha_n)$ and $X$ are both tangent to $C$ and hence their Lie bracket is also tangent to C. Thus its pairing with $\alpha_k$ vanish on $C$.

Note that $\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \alpha_{n-1})\big|_C$ is tangent to $C$ and $\alpha_n \big|_C \in (TC)^0.$ As a consequence, the function $$\begin{aligned}
\Pi (\alpha_1,\ldots, \alpha_n) = \langle \alpha_n, \Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \alpha_{n-1})\rangle\end{aligned}$$ is zero on $C.$ Therefore, the differential $d (\Pi(\alpha_1, \ldots, \alpha_n))$ restricted to $C$ is in $(TC)^0$, which in turn implies that the second term of the right hand side also vanish on $C$. Hence $$\begin{aligned}
 [\alpha_1, \ldots, \alpha_n]\big|_C \in (TC)^0.\end{aligned}$$ One can check that the restriction to $C$ does not depend on the chosen extension. Hence it defines a bracket on the sections of the conormal bundle $(TC)^0 \rightarrow C$.

1.  Let $m_0 \in M$ such that $\Pi (m_0) = 0$. Then $\{m_0\}$ is a coisotropic submanifold of $M$. In this case, the conormal structure becomes $T_{m_0}^*M$, which is a Filippov algebra.

2.  The Nambu structure of a Nambu-Lie group G vanishes at the identity element and therefore the dual $\mathfrak{g}^*$ of the Lie algebra $\mathfrak{g}$ of G has a Filippov algebra structure [@vais].

\[nlg-bracket-anchor\] Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$. By the Proposition \[coiso-n-bracket\], we see that the space of sections of the conormal bundle $ A^*G = (TM)^0 \rightarrow M$ admits a skew-symmetric $n$-bracket $[~,\ldots,~]$ and there exists a bundle map $$\rho := \Pi^{\sharp}\big|_{{\bigwedge}^{n-1} (TM)^0} : {\bigwedge}^{n-1} A^*G = {\bigwedge}^{n-1} (TM)^0 \rightarrow TM ,$$ as $M$ is a coisotropic submanifold of $G$.

Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$. Let $f \in C^\infty(M).$ Then by part ($5$) of the Theorem \[multiplicative\], $\iota_{d(\beta^*f)} \Pi$ is a left invariant $(n-1)$-vector field on $G$. Therefore, there exists an $(n-1)$-multisection $\delta^0_\Pi (f)\in \Gamma{\bigwedge^{n-1}AG}$ of the Lie algebroid $AG$ such that $$\begin{aligned}
 \iota_{d(\beta^*f)} \Pi = \overleftarrow{\delta^0_\Pi (f)} .\end{aligned}$$ Then we have the following result.

Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid of order $n$ and $AG \rightarrow M$ be its Lie algebroid. Then for any $ X \in \Gamma AG$, $$\begin{aligned}
 \mathcal{L}_{\overleftarrow{X}} \Pi := [\overleftarrow{X}, \Pi]\end{aligned}$$ is a left invariant $n$ vector field on $G$, where $\overleftarrow{X}$ is the left invariant vector field on $G$ corresponding to $X$. Moreover $\mathcal{L}_{\overleftarrow{X}} \Pi$ corresponds to the $n$-multisection $- \delta^1_\Pi (X) \in \Gamma{\bigwedge^n AG}$, that is, $$\begin{aligned}
 \mathcal{L}_{\overleftarrow{X}} \Pi = -\overleftarrow{\delta^1_\Pi (X)}\end{aligned}$$ where $\delta^1_\Pi (X) \in \Gamma{\bigwedge^n AG}$ is given by $$\begin{aligned}
\delta^1_\Pi (X) (\alpha_1, \ldots, \alpha_n) = \sum_{k=1}^{n} (-1)^{n-k} \Pi^{\sharp} (\alpha_1 \wedge \cdots \wedge \hat{\alpha}_k \wedge \cdots \wedge \alpha_n) (X(\alpha_k))
- X([\alpha_1,\ldots, \alpha_n])\end{aligned}$$ for $\alpha_1, \ldots, \alpha_n \in \Gamma A^*G = \Gamma (TM)^0.$

Let $\mathcal{X}_t = \text{exp} \hspace*{0.05cm} tX$ be the one-parameter family of bisections generated by $X \in \Gamma AG.$ Let $g \in G$ with $\beta(g) = u$. Let $u_t = (\text{exp} \hspace*{0.05cm} tX) (u)$ be the integral curve of $\overleftarrow{X}$ starting from $u$. If $\mathcal{G}$ is any (local) bisection through $g$, then from the multiplicativity condition of $\Pi$ (cf. Theorem \[multiplicative\]), we have $$\begin{aligned}
 \Pi (  g u_t) =   (r_{\mathcal{X}_t})_* \Pi(g) + (l_{\mathcal{G}})_* \Pi(u_t)-  (r_{\mathcal{X}_t})_* (l_{\mathcal{G}})_* \Pi(u).\end{aligned}$$ Therefore, $$\begin{aligned}
(r_{\mathcal{X}^{-1}_t})_* \Pi ( g u_t) - \Pi(g) =  (r_{\mathcal{X}^{-1}_t})_*   (l_{ \mathcal{G}})_* \Pi(u_t) - (l_{\mathcal{G}})_* \Pi (u).\end{aligned}$$ Taking derivative at $t = 0$, one obtains $$\begin{aligned}
 (\mathcal{L}_{\overleftarrow{X}} \Pi)(g) = (l_{\mathcal{G}})_* ((\mathcal{L}_{\overleftarrow{X}} \Pi)(u)).\end{aligned}$$ Therefore, $\mathcal{L}_{\overleftarrow{X}} \Pi $ is left invariant by the Proposition \[left-invariant\] and hence it corresponds to some $n$-multisection of $AG$. To show that $\mathcal{L}_{\overleftarrow{X}} \Pi$ corresponds to $- \delta^1_\Pi (X) \in \Gamma {\bigwedge^n AG}$, we have to check that $\mathcal{L}_{\overleftarrow{X}} \Pi$ and $- \overleftarrow{\delta^1_\Pi (X)}$ coincide on the unit space $M$ (both being left invariant). Since both of them are tangent to $\alpha$-fibers, it is enough to show that they coincide on the conormal bundle $(TM)^0$. Let $\alpha_1, \ldots, \alpha_n$ be any sections of $(TM)^0$ and $\tilde{\alpha}_1, \ldots, \tilde{\alpha}_n$ be their respective extensions to one forms on $G$. Observe that $$\begin{aligned}
& (\mathcal{L}_{\overleftarrow{X}} \Pi)\big|_M (\alpha_1, \ldots, \alpha_n)\\
 =& \big[\langle \overleftarrow{X}, d (\Pi(\tilde{\alpha}_1,\ldots,\tilde{\alpha}_n)) \rangle - \sum_{k=1}^{n} \Pi(\tilde{\alpha}_1 ,\ldots, \mathcal{L}_{\overleftarrow{X}} \tilde{\alpha}_k,\ldots, \tilde{\alpha}_n)\big]\big|_M\\
 =& \big[\langle \overleftarrow{X}, [\tilde{\alpha}_1,\ldots,\tilde{\alpha}_n] \rangle - \sum_{k=1}^n (-1)^{n-k} \langle \overleftarrow{X}, \iota_{\Pi^{\sharp} (\tilde{\alpha}_1 \wedge\cdots \wedge \hat{\tilde{\alpha}}_k \wedge\cdots \wedge \tilde{\alpha}_n)} d\tilde{\alpha}_k \rangle\\
  & - \sum_{k=1}^{n} (-1)^{n-k} \langle \Pi^{\sharp} (\tilde{\alpha}_1 \wedge\cdots \wedge \hat{\tilde{\alpha}}_k \wedge\cdots \wedge \tilde{\alpha}_n), \mathcal{L}_{\overleftarrow{X}} \tilde{\alpha}_k \rangle \big]\big|_M\\
  & \hspace*{5cm} (\text{from the Equation}~ (\ref{nambu-bracket}))\\
 =& \big[\langle \overleftarrow{X}, [\tilde{\alpha}_1,\ldots,\tilde{\alpha}_n] \rangle - \sum_{k=1}^n (-1)^{n-k} \langle \Pi^{\sharp} (\tilde{\alpha}_1 \wedge\cdots \wedge \hat{\tilde{\alpha}}_k \wedge\cdots \wedge \tilde{\alpha}_n), d \iota_{\overleftarrow{X}} \tilde {\alpha}_k \rangle \big]\big|_M\\
  & \hspace*{5cm} (\text{using Cartan formula})\\
 =& \langle X, [\alpha_1,\ldots,\alpha_n] \rangle - \sum_{k=1}^n (-1)^{n-k} \Pi^{\sharp} ({\alpha}_1 \wedge\cdots \wedge \hat{\alpha}_k \wedge\cdots \wedge {\alpha_n}) (X (\alpha_k))\\
 = & -\delta^1_\Pi (X) (\alpha_1,\ldots, \alpha_n).\end{aligned}$$

To make our notation simple, let us denote $\delta^0_\Pi, \delta^1_\Pi$ by the same symbol $\delta_\Pi$. We extend $\delta_\Pi$ to the graded algebra $\Gamma{\bigwedge^{\bullet}A}$ of multisections of $AG$ by the following rule $$\begin{aligned}
 \delta_\Pi (P \wedge Q) = \delta_\Pi (P) \wedge Q + (-1)^{|P| (n-1)} P \wedge \delta_\Pi (Q)\end{aligned}$$ for $P \in \Gamma{\bigwedge^{|P|}A}, Q \in \Gamma{\bigwedge^{|Q|}A}$. Then the operator $$\begin{aligned}
 \delta_\Pi : \Gamma{{\bigwedge}^kAG} \rightarrow \Gamma{{\bigwedge}^{k+n -1} AG}\end{aligned}$$ satisfies $$\begin{aligned}
 \delta_\Pi ([P,Q]) = [ \delta_\Pi (P), Q] + (-1)^{(|P|-1)(n-1)} [P, \delta_\Pi (Q)].\end{aligned}$$ Note that the operator $\delta_\Pi$ need not satisfy condition $\delta_\Pi \circ \delta_\Pi = 0 .$

We known that, Lie bialgebroids are infinitesimal form of Poisson groupoids. More precisely, given a Poisson groupoid $G \rightrightarrows M$ with Lie algebroid $AG$, it is known that its dual bundle $A^*G$ also carries a Lie algebroid structure and $(AG, A^*G)$ forms a Lie bialgebroid. In the next theorem we show that weak Lie-Filippov bialgebroids are infinitesimal form of Nambu-Lie groupoids.

\[nambu-grpd-bialgbd\] Let $(G \rightrightarrows M , \Pi) $ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \rightarrow M$. Then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ over $M$.

From the Remark \[nlg-bracket-anchor\], we have the space of sections of the bundle $A^*G = (TM)^0 \rightarrow M$ admits a skew-symmetric $n$-bracket $[~, \ldots, ~]$ and there exists a bundle map $$\rho : {\bigwedge}^{n-1}A^*G \rightarrow TM.$$

Let $\alpha_1, \ldots, \alpha_{n-1} \in \Gamma (TM)^0=\Gamma{(A^*G)} $ with $d_A \alpha_i = 0$, for all $i= 1, \ldots ,n-1.$ Let $\tilde{\alpha}_1, \ldots, \tilde{\alpha}_{n-1}$ be, respectively, their extensions to left invariant $1$-forms on $G$ such that $d \tilde{\alpha}_i = 0$, for all $i= 1, \ldots ,n-1.$ Then the conditions $(2)$ and $(3)$ of the Definition \[lie-fill-defn\] of a weak Lie-Filippov algebroid pair follows from the weak Lie-Filippov bialgebroid structure $(TG, T^*G)$ (Note that $G$ is a Nambu-Poisson manifold).

Let $f \in C^\infty(M).$ Then observe that $$\begin{aligned}
 [\tilde {\alpha}_1, \ldots, \tilde{\alpha}_{n-1}, (\beta^*f) \tilde{\alpha}_n ] = (\beta^*f) [\tilde{\alpha}_1, \ldots, \tilde{\alpha}_{n-1},  \tilde{\alpha}_n ] + \Pi^{\sharp}(\tilde{\alpha}_1 \wedge \cdots \wedge \tilde{\alpha}_{n-1})(\beta^*f)\tilde{\alpha}_n.\end{aligned}$$ Since $(\beta^*f) \tilde{\alpha}_n = \widetilde{f \alpha}_n$, by assertion $(2)$ of the Propostion \[coiso-n-bracket\], we get $$\begin{aligned}
[\alpha_1,\ldots, \alpha_{n-1}, f \alpha_n] = f [\alpha_1, \ldots, \alpha_n] + \rho (\alpha_1 \wedge \cdots \wedge \alpha_{n-1}) (f) \alpha_n\end{aligned}$$ proving condition (4) of the Definition \[lie-fill-defn\]. Moreover the compatibility condition of the weak Lie-Filippov bialgebroid (condition $(5)$ of the Definition \[lie-fill-defn\]) follows from the observation that for any $\alpha \in \Gamma{(A^*G)} = \Gamma (TM)^0$ and any left invariant extension $\tilde{\alpha} \in \Omega^1(G)$, we have $$\begin{aligned}
 d_A \alpha = (d \tilde \alpha)|_M.\end{aligned}$$ Thus, $(AG, A^*G)$ is a weak Lie-Filippov bialgebroid of order $n$.

If $(G, \Pi)$ is a Nambu-Lie group with Lie algebra $\mathfrak{g}$, the dual vector space $\mathfrak{g}^*$ carries a Filippov algebra structure [@vais]. Moreover the pair $(\mathfrak{g} , \mathfrak{g}^*)$ forms a (weak) Lie-Filippov bialgebra ([@bas-bas-das-muk; @vais]). The Lie-Filippov bialgebra $(\mathfrak{g}, \mathfrak{g}^*)$ is the infinitesimal form of the Nambu-Lie group $(G, \Pi).$ A Lie-Filippov bialgebra $(\mathfrak{g}, \mathfrak{g}^*)$ can also be seen a Lie algebra $\mathfrak{g}$ together with a Filippov algebra structure on the dual vector space $\mathfrak{g}^*$ such that the map $\delta : \mathfrak{g} \rightarrow \bigwedge^n \mathfrak{g}$ dual to the Filippov bracket on $\mathfrak{g}^*$, defines a $1$-cocycle of $\mathfrak{g}$ with respect to the adjoint representation on $\bigwedge^n \mathfrak{g}$.

We have seen that given a Nambu-Lie groupoid of order $n$, there is an induced Nambu-Poisson structure on the base manifold (cf. Proposition \[inverse-basenambu\]). On the other hand, given a weak Lie-Filippov bialgebroid, there is an induced Nambu-Poisson structure on the base (cf. Theorem \[wlfb-base-nambu\]). The next proposition compares these Nambu-Poisson structures on the base induced from the Nambu Lie groupoid and its infinitesimal.

\[compare-two-NP-structure\] Let $(G \rightrightarrows M , \Pi) $ be a Nambu-Lie groupoid (of order $n$) with associated weak Lie-Filippov bialgebroid $(AG, A^*G).$ Then the induced Nambu structures on $M$ coming from the Nambu-Lie groupoid and the weak Lie-Filippov bialgebroid are related by $$\begin{aligned}
 \{ ~, \ldots, ~\}_M  = (-1)^{n-1}\{ ~, \ldots, ~\}_{(AG, A^*G)} .\end{aligned}$$

For any functions $f_1, \ldots, f_n \in C^\infty(M)$, we have $$\begin{aligned}
 \{f_1, \ldots, f_n\}_{(AG, A^*G)} =& \Pi^{\sharp}\big|_M (d_A f_1 \wedge \cdots \wedge d_A f_{n-1}) f_n\\
=& \Pi^{\sharp} ( d (\beta^* f_1) \wedge \cdots \wedge d (\beta^* f_{n-1}))\big|_M f_n\\
=& \Pi (\beta^* f_1 , \ldots, \beta^* f_{n-1}, \beta^* f_n)\big|_M\\
=& (-1)^{n-1} \big(  \beta^*\{f_1, \ldots, f_n\} \big) \big|_M\\
=& (-1)^{n-1} \{f_1, \ldots, f_n\}_M.\end{aligned}$$

It is known that under some connectedness and simply connectedness assumption, any Lie bialgebra integrates to a Poisson-Lie group [@lu-wein], and any Lie bialgebroid integrates to a Poisson groupoid [@mac-xu2]. These results does not hold in the context of Nambu structures of order $\geqslant 3$. Let $G$ be a connected and simply-connected Lie group with Lie algebra $\mathfrak{g}.$ Given a Lie-Filippov bialgebra structure $(\mathfrak{g}, \mathfrak{g}^*)$ on $\mathfrak{g}$, the $1$-cocycle $\delta : \mathfrak{g} \rightarrow \bigwedge^n \mathfrak{g}$ dual to the Filippov algebra bracket on $\mathfrak{g}^*$ integrates a multiplicative $n$-vector field $\Pi$ on the Lie group. However this $n$-vector field (for $n \geqslant 3$) need not be a Nambu tensor [@vais], that is, need not be locally decomposable. Thus (weak) Lie-Filippov bialgebra does not integrate to a Nambu-Lie group in general.

Coisotropic subgroupoids of a Nambu-Lie groupoid
================================================

In this final section, we introduce the notion of coisotropic subgroupoid of a Nambu-Lie groupoid and study the infinitesimal object corresponding to it.

Let $(G \rightrightarrows M , \Pi)$ be a Nambu-Lie groupoid of order $n$. Then a subgroupoid $H \rightrightarrows N$ is called a [*coisotropic subgroupoid*]{} if $H$ is a coisotropic submanifold of $G$ with respect to $\Pi$.

1.  For $n=2$, that is, when $G \rightrightarrows M$ is a Poisson groupoid, this notion is same as the coisotropic subgroupoid of a Poisson groupoid introduced in [@xu].

2.  Let $(G, \Pi)$ be a Nambu-Lie group. Then a subgroup of $G$ is called coisotropic if it is also a coisotropic submanifold of $G$. Any coisotropic subgroup of $G$ is a coisotropic subgroupoid over a point.

3.  Let $(G \rightrightarrows M , \Pi)$ be a Nambu-Lie groupoid. Then by the Proposition \[inverse-basenambu\], there exist an induced Nambu-structure on $M$ for which the source map $\alpha$ is a Nambu-Poisson map. Let $N \hookrightarrow M$ be a coisotropic submanifold of $M$ with respect to this induced Nambu structure. Consider the restriction $G\big|_N := \alpha^{-1}(N) \cap \beta^{-1}(N)$, then $G\big|_N \rightrightarrows N$ is a coisotropic subgroupoid.

4.  Let $G \rightrightarrows M$ be a Nambu-Lie groupoid. If the set of all elements of $G$ which has same source and target, is a submanifold of $G$, then it is a coisotropic subgroupoid.

Note that, the infinitesimal object corresponding to a Nambu-Lie groupoid $(G\rightrightarrows M , \Pi)$ is the weak Lie-Filippov bialgebroid $(AG, A^*G)$. Therefore it is natural to ask how the Lie algebroid of a coisotropic subgroupoid $H \rightrightarrows N$ is related to the weak Lie-Filippov bialgebroid $(AG, A^*G)$. To answer this question, we introduce a notion of [*coisotropic subalgebroid*]{} of a weak Lie-Filippov bialgebroid and show that infinitesimal forms of coisotropic subgroupoids of a Nambu-Lie groupoid $(G \rightrightarrows M, \Pi)$ appear as coisotropic subalgebroids of the corresponding weak Lie-Filippov bialgebroid $(AG, A^*G)$.

\[coiso-subalg\] Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid of order $n$ over $M$. Then a Lie subalgebroid $B \rightarrow N$ of $A\rightarrow M$ is called a [*coisotropic subalgebroid*]{} if the anchor $\rho: \bigwedge^{n-1}A^* \rightarrow TM$ and the $n$-bracket $[~, \ldots, ~]$ on $\Gamma{A^*}$ satisfy the following properties.

1.  The anchor $\rho$ maps $\bigwedge^{n-1}B^{0} \rightarrow TN$.

2.  If $\alpha_1, \ldots, \alpha_n \in \Gamma{A^*}$ with ${\alpha_i}\big|_N \in B^{0}$ for all $i$, then $[\alpha_1, \ldots, \alpha_n]\big|_N \in B^{0}$.

3.  If $\alpha_1, \ldots, \alpha_n \in \Gamma{A^*}$ with ${\alpha_i}\big|_N \in B^{0}$ for all $i$ and $\alpha_n\big|_N = 0$, then $[\alpha_1, \ldots, \alpha_n]\big|_N = 0,$ where $B_x^{0} = \{\gamma \in A_x^* | \gamma (v) = 0, \forall v \in B_x\}$, is the annihilator of $B_x$, $x \in N$.

Let $M$ be a Nambu-Poisson manifold, then $(TM, T^*M)$ is a weak Lie-Filippov bialgebroid over $M$. Let $N \hookrightarrow M$ be a coisotropic submanifold. Then from the Proposition \[coiso-n-bracket\], it follows that the tangent bundle $TN \rightarrow N$ is a coisotropic subalgebroid.

It is known that (Proposition \[wlfb-base-nambu\], see also [@bas-bas-das-muk]), the base of a weak Lie-Filippov bialgebroid carries a Nambu structure. The next Proposition shows that the base of a coisotropic subalgebroid is a coisotropic submanifold with respect to this induced Nambu structure.

Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid over $M$ and $B \rightarrow N$ be a coisotropic subalgebroid. Then $N$ is a coisotropic submanifold of $M$.

Let $a : A \rightarrow TM$ denote the anchor of the Lie algebroid $A$ and $\rho : \bigwedge^{n-1}A^* \rightarrow TM$ be the anchor of pair $(A, A^*).$ We first show that, $a^* (TN)^0 \subseteq B^{0}$. This is true because, $ \langle a^*\xi_x, v \rangle = \langle \xi_x, a(v)\rangle = 0$ for $\xi_x \in (TN)_x^0$ and $v \in B_x$.

Let $\Pi_{(A, A^*)}$ be the induced Nambu structure on $M$ coming from the weak Lie-Filippov bialgebroid $(A, A^*).$ Then the induced map $\Pi^{\sharp}_{(A, A^*)} : \bigwedge^{n-1}T^*M \rightarrow TM$ is given by $\Pi^{\sharp}_{(A, A^*)} = \rho \circ \bigwedge^{n-1}a^*$. Therefore, for any $\xi_1, \ldots, \xi_{n-1} \in (TN)^0$, we have $$\Pi^{\sharp}_{(A, A^*)} (\xi_1, \ldots, \xi_{n-1}) = \rho (a^*\xi_1, \ldots, a^*\xi_{n-1}) \in TN$$ as $a^*\xi_i \in B^{0}$ and $B$ is a coisotropic subalgebroid. Therefore $N$ is a coisotropic submanifold of $M$.

The next proposition shows that the infinitesimal object corresponding to coisotropic subgroupoids are coisotropic subalgebroids.

\[coiso-subgrpd-coiso-subalgbd\] Let $(G \rightrightarrows M, \Pi)$ be a Nambu-Lie groupoid with weak Lie-Filippov bialgebroid $(AG, A^*G)$. Let $H \rightrightarrows N$ be a coisotropic subgroupoid of $G \rightrightarrows M$ with Lie algebroid $AH \rightarrow N$. Then $AH \rightarrow N$ is a coisotropic subalgebroid.

Since $H \rightrightarrows N$ is a Lie subgroupoid of $G \rightrightarrows M$, therefore $AH \rightarrow N$ is a Lie subalgebroid of $AG \rightarrow M$. We claim that the anchor $\rho = \Pi^{\sharp}\big|_{\bigwedge^{n-1}(TM)^0} = \Pi^{\sharp}\big|_{\bigwedge^{n-1} (A^*G)}$ of the weak Lie-Filippov bialgebroid $(AG, A^*G)$ maps $\bigwedge^{n-1}(AH)^0$ to $TN$. First observe that, for any $x \in N$, $(AH)_x^0 = (TM)_x^0 \cap (TH)_x^0$ and $T_xN = T_xM \cap T_xH$. Therefore, $\rho$ maps $\bigwedge^{n-1} (AH)^0$ to $$\Pi^{\sharp}({\bigwedge}^{n-1} (TM)^0) \cap \Pi^{\sharp}({\bigwedge}^{n-1} (TH)^0) \subseteq TM \cap TH \cong TN,$$ here we have used the fact that $M$ and $H$ are both coisotropic submanifolds of $G$.

Let $\alpha_1, \ldots, \alpha_n \in \Gamma{A^*G} = \Gamma{(TM)^0}$ such that $\alpha_i\big|_N \in (AH)^0$, for all $i=1, \ldots, n$. Let $\tilde{\alpha}_1, \ldots, \tilde{\alpha}_n$ be one forms on $G$ extending $\alpha_1, \ldots, \alpha_n$ and are conormal to $H$. Then by the Proposition \[coiso-n-bracket\], the $1$-form $[\tilde{\alpha_1}, \ldots, \tilde{\alpha_n}]$ is conormal to both $M$ and $H$, as $M$ and $H$ are both coisotropic submanifolds of $G$. Therefore, $$[\tilde{\alpha_1}, \ldots, \tilde{\alpha_n}]\big|_N \in (TM)^0 \cap (TH)^0 \cong (AH)^0.$$ Verification of the last condition of the Definition \[coiso-subalg\] is similar. Hence $AH \rightarrow N$ is a coisotropic subalgebroid of $(AG, A^*G).$

Let $(G \rightrightarrows M , \Pi)$ be a Nambu-Lie groupoid and $H \rightrightarrows N$ be a coisotropic subgroupoid. Then $N$ is a coisotropic submanifold of $M$.

[BFGM03]{}

Samik Basu, Somnath Basu, Apurba Das and Goutam Mukherjee, [[*Nambu structures and associated Bialgebroids*]{}]{}, preprint, arXiv:1502.06533

Jean-Paul Dufour and Nguyen Tien Zung, [[*Poisson structures and their normal forms*]{}]{}, Birkh$\ddot{\text{a}}$user Verlag, Basel-Boston-Berlin (2005).

V. T. Filippov, [[*n-Lie algebras*]{}]{}, Sib. Math. Zh., [**26 (6)**]{}, 126-140, (1985).

J. Grabowski and G. Marmo, [[*On Filippov algebroid and multiplicative Nambu structures*]{}]{}, Diff. Geom. Appl. [**12**]{} (2000) 35-50.

R. Ib$\acute{\text{a}}\tilde{\text{n}}$ez, M. de Le$\acute{\text{o}}$n, J.C. Marrero, E. Padr$\acute{\text{o}}$n, [[*Leibniz algebroid associated with a Nambu-Poisson structure*]{}]{}, J. Phys. A: Math. Gen. [**32**]{} (1999) 8129-8144.

R. Ib$\acute{\text{a}}\tilde{\text{n}}$ez, M. de Le$\acute{\text{o}}$n , J.C. Marrero and D. M. de Diego, [[*Reduction of generalized Poisson and Nambu-Poisson manifolds*]{}]{}, Rep. Math. Phys. [**42**]{} (1998) 71-90.

Y. Kosmann-Schwarzbach; [[*Exact Gerstenhaber algebras and Lie bialgebroids*]{}]{}, Acta applicandae Mathematicae [[**41**]{}]{}: 153-165, 1995.

Jiang-Hua Lu and Alan Weinstein, [[*Poisson Lie groups, dressing transformations, and Bruhat decompositions*]{}]{}, J. Diff. Geom. [**31**]{} (1990) 501-526.

Kirill C.H. Mackenzie, [[*Lie groupoids and Lie algebroids in Differential Geometry*]{}]{}, Cambridge University Press, Cambridge, (2005).

Kirill C.H. Mackenzie and Ping Xu, [[*Lie bialgebroids and Poisson groupoids*]{}]{}, Duke Math. J. [**73**]{}, (1994) 415-452.

Mackenzie, Kirill C. H., and Xu, P.: [[*Integration of Lie bialgebroids*]{}]{}, Topology [**39**]{} (2000) 445-467.

Yoichiro Nambu, [[*Generalized Hamiltonian dynamics*]{}]{}, Phys. Rev. D, [**7**]{} (1973) 2405-2412.

David Iglesias Ponte, Camille Laurent-Gengoux and Ping Xu, [[*Universal lifting theorem and quasi-Poisson groupoids*]{}]{}, J. Eur. Math. Soc. (JEMS) [**14**]{} (2012), no. 3, 681-731.

Leon Takhtajan, [*On foundation of the Generalized Nambu Mechanics*]{}, Comm. Math. Phys. Volume [**160**]{}, Number 2 (1994), 295-315.

Izu Vaisman, [[*Lectures on the Geometry of Poisson Manifolds*]{}]{}, Progress in Mathematics, Vol. 118, Birkh$\ddot{\text{a}}$user Verlag, Basel, (1994).

Izu Vaisman, [[*Nambu-Lie groups*]{}]{}, J. Lie Theory [**10**]{} (2000), 181-194.

Alan Weinstein, [[*Coisotropic calculus and Poisson groupoids*]{}]{}, J. Math. Soc. Japan [**40**]{} (1988) 705-727.

Ping Xu, [[*On Poisson groupoids*]{}]{}, Inter. J. math. [**6**]{}, No. 1, 101-124 (1995).
