1 Introduction and notation

Definition 2.14 .

Given a $\mathbb{Z}_{2}$ -graded algebra $A=A_{\bar{0}}\oplus A_{\bar{1}}$ , (i.e., $A_{\varepsilon}\cdot A_{\delta}\subset A_{\varepsilon+\delta}$ for any $\varepsilon,\delta\in\mathbb{Z}_{2}=\mathbb{Z}/(2)=\{\overline{0},\overline{1}\}$ ), we say that $A$ is a commutative superalgebra if

ba=(-1)^{\varepsilon\delta}ab

for any $a\in A_{\varepsilon}$ and $b\in A_{\delta}$ , $\varepsilon,\delta\in\mathbb{Z}_{2}=\{\bar{0},\bar{1}\}$ .

Definition 2.15 .

A bracketed superalgebra is a pair $(A,\{\cdot,\cdot\})$ where $A$ is a commutative superalgebra and a bilinear map $\{\cdot,\cdot\}:A\times A\to A$ such that $\{A_{\varepsilon},A_{\delta}\}\subset A_{\varepsilon+\delta}$ for any $\varepsilon,\delta\in\mathbb{Z}_{2}$ and the following identities hold:

(i) super-anti-commutativity :

\{a,b\}+(-1)^{\varepsilon\delta}\{b,a\}=0

for any $a\in A_{\varepsilon},b\in A_{\delta}$ ,

(ii) the super-Leibniz rule

(2.4)

\{a,bc\}=\{a,b\}c+(-1)^{\varepsilon\delta}b\{a,c\}

for any $a\in A_{\varepsilon},b\in A_{\delta},c\in A_{\gamma}$ .

If, in addition, the bracket satisfies

(iii) the super-Jacobi identity :

(-1)^{\varepsilon\gamma}\{a,\{b,c\}\}+(-1)^{\gamma\delta}\{c,\{a,b\}\}+(-1)^{% \delta\varepsilon}\{b,\{c,a\}\}=0

for any $a\in A_{\varepsilon},b\in A_{\delta},c\in A_{\gamma}$ , then we will refer to $A$ as a Poisson superalgebra and will refer to $\{\cdot,\cdot\}$ as super-Poisson bracket .

Définition 2.2

Soit $(N,\delta)$ un espace de longueur de diamètre inférieur à $\pi$ , on appelle sinus produit tordu, l’espace de longueur $((0,\pi)\times N,d)$ où la distance $d$ est définie pour $(t,x),(s,y)$ dans $(0,\pi)\times N$ , par

\cos d((t,x),(s,y))=\cos s\cos t+\sin s\sin t\cos\delta(x,y).

(11)

Definition 5.2.2

( The $d^{\prime}$ and $d^{\prime\prime}$ Laplacians )

\framebox{$d^{\prime\star}=-\star d^{\prime\prime}\star,$}\qquad\framebox{$d^{% \prime\prime\star}=(d^{\prime\prime})^{\star}=-\star d^{\prime}\star$}

\framebox{$\Delta^{\prime}=d^{\prime}d^{\prime\star}+d^{\prime\star}d^{\prime}% ,$}\qquad\framebox{$\Delta^{\prime\prime}=d^{\prime\prime}d^{\prime\prime\star% }+d^{\prime\prime\star}d^{\prime\prime}.$}

Definition 3.3 .

Take $\lambda\in\Lambda_{n}$ and $\mu\in\Lambda_{m}$ with $\lambda\cap\mu=\emptyset$ . For $t\in l(\lambda)$ and $s\in l(\mu)$ , we define $ts\in l(\lambda\cup\mu)$ by

(ts)(i)=\begin{cases}t(i)&\text{for $i=1,\ldots,n$}\\ s(i-n)&\text{for $i=n+1,\ldots,n+m$}.\end{cases}

Definition 12 .

A family of diffeomorphisms $\phi^{\varepsilon}:\mathcal{M}\rightarrow\mathcal{M}$ is called fiber-preserving if there exists a family of diffeomorphisms on the base space $\varphi^{\varepsilon}:\mathcal{B}\rightarrow\mathcal{B}$ such that

\pi\circ\phi^{\varepsilon}=\varphi^{\varepsilon}\circ\pi.

We say that $\phi^{\varepsilon}$ is vertical whenever $\varphi^{\varepsilon}=\mathbb{I}$ for all $\varepsilon$ .

Definition 2.1 .

Let $H$ be a Lie group and $G$ a Lie subgroup of $H$ . Denote by $\mathfrak{h}$ the Lie algebra of $H$ and by $\mathfrak{g}$ the Lie algebra of $G$ . We shall say that $G$ is a reductive Lie subgroup of $H$ if there exists a direct sum decomposition

\mathfrak{h}=\mathfrak{g}\oplus\mathfrak{m},

where $\mathfrak{m}$ is an $\mathrm{Ad}_{G}$ -invariant vector subspace of $\mathfrak{h}$ , i.e. $\mathrm{Ad}_{a}(\mathfrak{m})\subset\mathfrak{m}$ for all $a\in G$ (which means that the $\mathrm{Ad}_{G}$ representation of $G$ in $\mathfrak{h}$ is reducible into a direct sum decomposition of two $\mathrm{Ad}_{G}$ -invariant vector spaces: cf. [ 23 ] , p. $83$ ).

Definition 2.7 .

Let $P(M,G)$ be a principal bundle and $\rho\colon\Gamma\to G$ a central homomorphism of a Lie group $\Gamma$ onto $G$ , i.e. such that its kernel is discrete and contained in the centre of $\Gamma$ [ 18 ] (see also [ 19 ] ). A $\Gamma$ -structure on $P(M,G)$ is a principal bundle map $\zeta\colon\tilde{P}\to P$ which is equivariant under the right actions of the structure groups, i.e.

\zeta(\tilde{u}\cdot\alpha)=\zeta(\tilde{u})\cdot\rho(\alpha)

for all $\tilde{u}\in\tilde{P}$ and $\alpha\in\Gamma$ .

Definition 1.6 ( [ A ] ) .

A braided groupoid is a collection $({\mathcal{V}},\rightharpoonup,\leftharpoonup)$ where ${\mathcal{V}}\rightrightarrows{\mathcal{P}}$ is a groupoid, $({\mathcal{V}},{\mathcal{V}},\rightharpoonup,\leftharpoonup)$ is a matched pair of groupoids and for every pair $(f,g)\in{\mathcal{V}}_{\mathfrak{e}}\times_{\mathfrak{s}}{\mathcal{V}}$ the following equation holds:

(1.9)

fg=(f\rightharpoonup g)(f\leftharpoonup g).

Definition 3.3.1 .

Define the twisted powers $\pi^{\{m\}}$ of $\pi$ by the two-way recurrence

\pi^{\{0\}}=1,\qquad\pi^{\{m+1\}}=(\pi^{\{m\}})^{\sigma}\pi.