\section{Introduction}

A degree $n$ symplectic $NQ$-manifold is an $\mathbb{N}$-graded manifold equipped with a degree $n$ symplectic structure and a symplectic homological vector field. There has recently been much interest in symplectic $NQ$-manifolds, primarily arising from the following one-to-one correspondences, due to Roytenberg \cite{royt:graded} and Severa \cite{severa:sometitle}:
\begin{align*}
\fbox{degree $1$ symplectic $NQ$-manifolds} &\longleftrightarrow \fbox{Poisson manifolds}\\
\fbox{degree $2$ symplectic $NQ$-manifolds} &\longleftrightarrow \fbox{Courant algebroids}
\end{align*}
These correspondences suggest that one should be able to study Poisson, Courant, and generalized complex geometries from the perspective of graded symplectic geometry. Following this premise, the papers \cite{bcmz} and \cite{cz:poisson} have shown that various constructions for Poisson, Courant, and generalized complex reduction can be interpreted as examples of symplectic reduction of $NQ$-manifolds. This point of view unifies the various methods of reduction and, in a sense, explains why they work.

Consider moment map reduction in the case where $\mathcal{S}$ is a symplectic $NQ$-manifold and $\mathcal{G}$ is a graded Lie group with a Hamiltonian action on $\mathcal{S}$. Under the usual regularity assumptions, a graded version of the Marsden-Weinstein theorem guarantees that the reduced space inherits a symplectic structure. In order to ensure that the homological vector field on $\mathcal{S}$ also descends to the quotient, one must impose additional hypotheses. In \cite{cz:poisson} and \cite{bcmz}, it was proposed (in the cases $n=1$ and $n=2$, respectively) that $\mathcal{G}$ should be allowed to have extra structure; namely, the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ should be a differential graded Lie algebra (DGLA). A DGLA structure on $\mathfrak{g}$ integrates to a multiplicative homological vector field on $\mathcal{G}$, making $\mathcal{G}$ into a \emph{$\mathrm{dg}$-group} or a \emph{$Q$-group}.  It was shown in \cite{cz:poisson} and \cite{bcmz} (again in their respective cases) that, if $\mathcal{G}$ is connected and the comoment map $\mathfrak{g}[n] \to C^\infty(\mathcal{S})$ is a morphism of DGLAs, then the vector field on $\mathcal{S}$ passes to the symplectic quotient.

On the other hand, the following example of Poisson quotients illustrates that $Q$-groups or DGLAs are insufficiently general to incorporate all known notions of reduction. Let $M$ be a Poisson manifold, and let $G$ be a Poisson Lie group with a Poisson action on $M$. Then it is known that the quotient $M/G$, if smooth (for example, if the action is free and proper), inherits a Poisson structure. This example may be restated in the language of graded symplectic geometry as follows. The degree $1$ symplectic $NQ$-manifold associated to the Poisson manifold $M$ is the shifted cotangent bundle $T^*[1]M$. The action of $G$ on $M$ naturally lifts to a Hamiltonian action on $T^*[1]M$, and the symplectic quotient may be canonically identified with $T^*[1](M/G)$. The homological vector field on $T^*[1]M$ naturally descends to $T^*[1](M/G)$, giving the Poisson structure on $M/G$.

The example that we have just described involves the action of a Poisson Lie group and does not fit into the framework of reduction by $Q$-group or DGLA action. This fact suggests that the theory of reduction of symplectic $NQ$-manifolds should allow the acting group to possess a general structure that includes both $Q$-groups and Poisson Lie groups as special cases. The search for such a generalization leads us to the notion of \emph{homotopy Poisson Lie group}.

A homotopy Poisson structure is an $L_\infty$ structure whose brackets satisfy a Leibniz rule. Such structures have appeared in the work of Voronov \cite{voronov:higher1,voronov:higher2} under the name ``higher Poisson brackets,''
and Cattaneo and Felder \cite{cf:formality} have called them $P_\infty$ structures. See \cite{bruce:higher} for more recent work involving such structures.

A homotopy Poisson Lie group is a graded Lie group whose algebra of functions has a multiplicative homotopy Poisson algebra structure. Both $Q$-groups and Poisson Lie groups are examples of homotopy Poisson Lie groups. The main result of this paper is a reduction result for homotopy Poisson Lie group actions---essentially, if a homotopy Poisson Lie group has a Hamiltonian action on a symplectic $NQ$-manifold, then under certain compatibility and regularity conditions, the symplectic quotient inherits a homological vector field. This result provides a quite general framework for moment map reduction of symplectic $NQ$-manifolds.

The structure of the paper is as follows. In \S\ref{sec:pinftyman}, we review the definition of homotopy Poisson manifolds and their characterization in terms of multivector fields. In \S\ref{sec:pinftylie}, we introduce homotopy Poisson Lie groups and describe their corresponding infinitesimal objects, which we call homotopy Lie bialgebras. We consider actions of homotopy Poisson Lie groups on homotopy Poisson manifolds in \S\ref{sec:pinftyact}, proving that if an action satisfies a compatibility condition, then its quotient inherits a homotopy Poisson structure. The heart of the paper is \S\ref{sec:hamiltonian}, where we turn to Hamiltonian actions of homotopy Poisson Lie groups on degree $1$ symplectic (not necessarily $N$)$Q$-manifolds. In \S\ref{sec:pinftyact2}, we show how the framework of \S\ref{sec:hamiltonian} includes homotopy Poisson quotients and, in particular, Poisson quotients. Finally, in \S\ref{sec:higher}, we describe a modification of the definition of homotopy Poisson Lie group that allows one to extend the results of \S\ref{sec:hamiltonian} to reduction of degree $n$ symplectic $Q$-manifolds.

\begin{ack}
 The author thanks Marco Zambon, Florian Sch\"{a}tz, Henrique Bursztyn, and Alberto Cattaneo for helpful comments and discussions.
\end{ack}

\section{Homotopy Poisson manifolds}\label{sec:pinftyman}
In this section, we recall the notions of homotopy Poisson algebra and homotopy Poisson manifold. For more details and applications of such structures, we refer the reader to \cite{cf:formality, cat:reduction, schatz:thesis} (under the name \emph{$P_\infty$-manifolds} and \cite{voronov:higher1, voronov:higher2, bruce:higher} (under the name \emph{higher Poisson manifolds}).

Let $\mathcal{A} = \sum_{i \in \mathbb{Z}} \mathcal{A}_i$ be a graded commutative algebra over a field $k$, and let $\tilde{\mathcal{A}} := \mathcal{A}[1]$ be the degree $1$ suspension of $\mathcal{A}$, so that $\tilde{\mathcal{A}}_i = \mathcal{A}_{i+1}$.
\begin{dfn}\label{dfn:palg}
A \emph{homotopy Poisson algebra structure} on $\mathcal{A}$ consists of a series of multilinear $\ell$-ary brackets $\beta_\ell : \tilde{\mathcal{A}}^{\otimes \ell} \to \tilde{\mathcal{A}}$ for $\ell \geq 0$ such that the following properties hold:
\begin{enumerate}
\item The degree of $\beta_\ell$ is $1$; that is, $|\beta_\ell(a_1, \dots, a_\ell)| = 1 + \sum |a_i|$ for all homogeneous $a_1, \dots, a_\ell \in \tilde{\mathcal{A}}$.
\item The brackets $\beta_\ell$ are graded symmetric.
\item If $a_1, \dots, a_{\ell-1}$ are homogeneous elements of $\tilde{\mathcal{A}}$, then $\beta_\ell(a_1, \dots, a_{\ell-1}, \cdot)$ is a graded derivation of degree $1 + \sum |a_i|$.
\item The brackets $\beta_\ell$ satisfy the generalized Jacobi identities (with the sign convention of \cite{voronov:higher1}; also see \cite{schatz:thesis}).
\end{enumerate}
\end{dfn}
In other words, a homotopy Poisson algebra is a graded commutative algebra with an $L_\infty$ structure such that the multibrackets satisfy a Leibniz rule. We may think of a homotopy Poisson structure as being a generalization of a Poisson structure, where the Jacobi identity is allowed to only hold ``up to homotopy.'' We remark, however, that homotopy Poisson structures do not give the cofibrant resolution of the Poisson operad, where the Leibniz rule, as well as the commutativity and associativity of multiplication, would be similarly weakened.

\begin{remarks}
\begin{enumerate}
\item The traditional definition of $L_\infty$-algebra does not include a $0$-ary bracket $\beta_0$. Following common terminology, we say that a homotopy Poisson algebra is \emph{flat} if $\beta_0$ vanishes.
\item If $\beta_\ell$ is thought of as a bracket on $\mathcal{A}$, as opposed to $\tilde{\mathcal{A}}$, then $\beta_\ell$ is of degree $2-\ell$.
\item We will say that a homotopy Poisson algebra is \emph{of finite type} if there exists a $q$ such that $\beta_\ell = 0$ for all $\ell > q$.
\end{enumerate}
\end{remarks}

Let $\mathcal{A}$ and $\mathcal{A}'$ be homotopy Poisson algebras.  A (strict) morphism of homotopy Poisson algebras from $\mathcal{A}$ to $\mathcal{A}'$ is a map $\eta: \mathcal{A} \to \mathcal{A}'$ such that $\eta(\beta_\ell(a_1, \dots, a_\ell)) = \beta^\prime_\ell(\eta(a_1), \dots, \eta(a_\ell))$ for all $a_1, \dots, a_\ell \in \mathcal{A}$. One could also try to define weak morphisms, but this will be unnecessary for the present purposes.

\begin{dfn}
A \emph{homotopy Poisson manifold} is a $\mathbb{Z}$-graded manifold $\mathcal{M}$ whose algebra of functions $C^\infty(\mathcal{M})$ is equipped with a homotopy Poisson algebra structure of finite type. A map $\psi:\mathcal{M} \to \mathcal{M}'$ of homotopy Poisson manifolds is a homotopy Poisson morphism if the pullback map $\psi^*: C^\infty(\mathcal{M}') \to C^\infty(\mathcal{M})$ is a morphism of homotopy Poisson algebras.
\end{dfn}

  Let $(\mathcal{M}, \beta_\ell)$ be a homotopy Poisson manifold.  Properties (1)-(3) in Definition \ref{dfn:palg} imply that for each $\ell$ there exists an $\ell$-vector field $\pi_\ell \in \mathfrak{X}^\ell (\mathcal{M})$ of degree $2 - \ell$ such that $\beta_\ell$ is given by the derived bracket formula
\begin{equation}\label{eqn:derived}
	\beta_\ell(f_1, \dots, f_\ell) = [[ \cdots [\pi_\ell, f_1], \cdots ], f_\ell]
\end{equation}
for $f_1, \dots, f_\ell \in C^\infty(\mathcal{M})$.

The algebra $\vect^\bullet(\mathcal{M})$ of multivector fields has a natural bigrading, and with respect to the total grading the $\ell$-vector fields $\pi_\ell$ are all of degree $2$.
If we let $\pi = \sum \pi_\ell$, then the generalized Jacobi identities are equivalent to the equation
\begin{equation}\label{eqn:integrability}
[\pi,\pi]=0.
\end{equation}
A degree $2$ multivector field satisfying \eqref{eqn:integrability} has been called a \emph{Poisson multivector field} by Sch\"{a}tz \cite{schatz:thesis}. The derived bracket \eqref{eqn:derived} relating Poisson multivector fields and homotopy Poisson structures is due to Voronov \cite{voronov:higher1, voronov:higher2}.

The following statement summarizes the above discussion.

\begin{prop}
 Let $\mathcal{M}$ be a $\mathbb{Z}$-graded manifold. The derived bracket formula \eqref{eqn:derived} gives a one-to-one correspondence between homotopy Poisson structures and Poisson multivector fields on $\mathcal{M}$.
\end{prop}

We leave the proof of the following proposition as an exercise for the reader.
\begin{prop}
If $\mathcal{M}$ and $\mathcal{M}'$ are homotopy Poisson manifolds with Poisson multivector fields $\pi$ and $\pi'$, respectively, then a smooth map $\psi: \mathcal{M} \to \mathcal{M}'$ is a homotopy Poisson morphism if and only if $\pi$ is $\psi$-related to $\pi'$.
\end{prop}

\begin{examples}
\begin{enumerate}
\item A homotopy Poisson manifold where $\beta_\ell = 0$ for $\ell \neq 1$ is a \emph{$Q$-manifold}, that is a graded manifold equipped with a degree $1$ vector field $\pi_1$ such that $(\pi_1)^2 = 0$; such a vector field is called \emph{homological}. Many interesting examples of $Q$-manifolds come from Lie algebroids \cite{vaintrob}.
\item A homotopy Poisson manifold where $\beta_\ell = 0$ for $\ell \neq 2$ is a graded Poisson manifold. In particular, a homotopy Poisson structure on an ordinary manifold is the same thing as a Poisson structure, since a degree $2$ multivector field on an ordinary manifold is necessarily a bivector field.
\item A homotopy Poisson manifold where $\beta_\ell$ is nonzero for only $\ell = 1,2$ is a \emph{$QP$-manifold}, that is a graded Poisson manifold equipped with a homological Poisson vector field. This type of structure appears, for example, in BRST quantization \cite{henneaux-teitelboim}.
\item By allowing $\beta_\ell$ to be nonzero for more than two values, one obtains more general possibilities. For example, if $\beta_\ell$ is nonzero for only $\ell = 0,1,2$, then equation \eqref{eqn:integrability} decomposes into the following set of properties:
\begin{itemize}
\item $\pi_2$ is a Poisson bivector.
\item $\pi_1$ is a Poisson vector field.
\item $\pi_0$ is a $\pi_1$-invariant function.
\item The failure of $\pi_1$ to be homological, $(\pi_1)^2$, equals the Hamiltonian vector field of $\pi_0$.
\end{itemize}
\end{enumerate}
\end{examples}

Let $\mathcal{M}$ be a homotopy Poisson manifold with Poisson multivector field $\pi$.  Define the operator $d_\pi := [\pi, \cdot]$ on $\vect^\bullet(\mathcal{M})$. This operator is of total degree $1$ since $\pi$ is of total degree $2$, and the integrability equation \eqref{eqn:integrability} holds if and only if $d_\pi^2 = 0$.

\begin{prop}\label{prop:correspondence}
 The map $\pi \mapsto d_\pi$ gives a bijection from Poisson multivector fields on a graded manifold $\mathcal{M}$ to degree $1$ differentials on $\vect^\bullet(\mathcal{M})$ that are derivations of the Schouten bracket.
\end{prop}

\begin{proof}
Let $\delta$ be a degree $1$ operator on $\vect^\bullet(\mathcal{M})$ that is a derivation of the Schouten bracket and such that $\delta^2=0$. Write $\delta = \sum \delta_\ell$, where $\delta_\ell$ takes elements of $\mathfrak{X}^i(\mathcal{M})$ to $\mathfrak{X}^{i + \ell -1}(\mathcal{M})$. For $\ell > 0$, we can then obtain an $\ell$-ary bracket $\beta^\delta_\ell$ by the formula
\begin{equation*}
 \beta^\delta_\ell(f_1, \dots, f_\ell) = [[ \cdots [\delta_\ell f_1, f_2], \cdots ], f_\ell].
\end{equation*}
The skew-symmetry and Leibniz rule for $\beta^\delta_\ell$ follow from the derivation property of $\delta_\ell$. If we define an $\ell$-vector field $\pi^\delta_\ell$ by the derived bracket equation
\begin{equation*}
	\beta^\delta_\ell(f_1, \dots, f_\ell) = [[ \cdots [[\pi^\delta_\ell, f_1],f_2], \cdots ], f_\ell],
\end{equation*}
then we immediately have $\delta_\ell = [\pi^\delta_\ell, \cdot]$.

To deal with the case $\ell = 0$, we use the Euler vector field $\varepsilon \in \mathfrak{X}(\mathcal{M})$ associated to the grading on $\mathcal{M}$, defined by $\varepsilon(f) = |f| f$ for a function $f \in C^\infty(\mathcal{M})$ of homogeneous degree $|f|$. Then we let $\pi^\delta_0 = -\frac{1}{2}\delta_0 \varepsilon$.

The reader may verify that the map $\delta \mapsto \pi^\delta$ that we have described is a two-sided inverse to the map $\pi \mapsto d_\pi$.
\end{proof}

Recall that, if $\mathcal{M}$ is a graded manifold, then $\vect^\bullet(\mathcal{M})$ is the algebra of polynomial functions on $T^*[1]\mathcal{M}$. From this point of view, the Schouten bracket of multivector fields can be identified with the Poisson bracket associated to the canonical degree $1$ symplectic structure on $T^*[1]\mathcal{M}$. Thus we may view Proposition \ref{prop:correspondence} as giving a one-to-one correspondence between $P_\infty$ structures on $\mathcal{M}$ and (polynomial) symplectic homological vector fields on $T^*[1]\mathcal{M}$.

\begin{remark}
 If $\mathcal{M}$ is a homotopy Poisson manifold, then $T^*[1]\mathcal{M}$ is a degree $1$ symplectic $Q$-manifold but in general will have coordinates in negative degrees even if $\mathcal{M}$ is nonnegatively graded. Therefore it is essential for the purposes of this paper that we consider symplectic $Q$-manifolds that are $\mathbb{Z}$-(as opposed to $\mathbb{N}$-)graded.
\end{remark}

\section{Homotopy Poisson Lie groups}\label{sec:pinftylie}
\begin{dfn}
A \emph{homotopy Poisson Lie group} is a graded Lie group $\mathcal{G}$ equipped with a homotopy Poisson manifold structure such that the multiplication map $m: \mathcal{G} \times \mathcal{G} \to \mathcal{G}$ is a homotopy Poisson morphism.
\end{dfn}
Examples of homotopy Poisson Lie groups include $Q$-groups (that is, graded Lie groups equipped with multiplicative homological vector fields) and Poisson Lie groups.

\begin{dfn} \label{dfn:leftinv}
Let $\mathcal{G}$ be a graded Lie group.  A multivector field $X \in \vect^\bullet(\mathcal{G})$ is \emph{left-invariant} if $(0,X) \in \vect^\bullet(\mathcal{G} \times \mathcal{G})$ is $m$-related to $X$; it is \emph{right-invariant} if $(X,0)$ is $m$-related to $X$.
\end{dfn}

The following statement is an analogue of the fact that for Poisson Lie groups, the Poisson bivector acts on left-invariant vector fields (see \cite{lu-wei}).
\begin{prop} \label{prop:pdgla}
 Let $\mathcal{G}$ be a homotopy Poisson Lie group with Poisson multivector field $\phi$.  If $X \in \vect^\bullet(\mathcal{G})$ is a left-invariant multivector field, then $[\phi,X]$ is also left-invariant; if $X$ is right-invariant, then $[\phi,X]$ is also right-invariant.
\end{prop}
\begin{proof}
By the definition of homotopy Poisson Lie group, we have that $(\phi,\phi)$ is $m$-related to $\phi$. Relatedness is preserved by the Schouten bracket, so if $X$ is left-invariant then $[(\phi,\phi), (0,X)]$ is $m$-related to $[\phi,X]$. On the other hand, $[(\phi,\phi), (0,X)] = ([\phi,0],[\phi,X]) = (0, [\phi,X])$. Therefore $[\phi,X]$ satisfies the left-invariance property of Definition \ref{dfn:leftinv}. The proof for right-invariant multivector fields is similar.
\end{proof}

It follows from Proposition \ref{prop:pdgla} that the differential $d_\phi := [\phi, \cdot]$ on $\vect^\bullet(\mathcal{G})$ may be restricted to the algebra of left-invariant multivector fields. If $\mathfrak{g}$ is the graded Lie algebra of $\mathcal{G}$, then we can identify $\bigS(\mathfrak{g} [-1])$ with the algebra of left-invariant multivector fields\footnote{We use $\bigS(\mathfrak{g} [-1])$ instead of $\bigwedge \mathfrak{g}$ in order to agree with our sign convention for multivector fields, where $\vect^\bullet(\mathcal{M}) = \bigS(\mathfrak{X}(\mathcal{M})[1])$.}. We denote the restriction of $d_\phi$ to $\bigS(\mathfrak{g} [-1])$ by $\hat{d}_\phi$. Thus we have the following infinitesimal description of homotopy Poisson Lie groups.

\begin{dfn}\label{dfn:pinftylie}
 A \emph{homotopy Lie bialgebra} is a graded Lie algebra $\mathfrak{g}$ equipped with a differential $\hat{d}_\phi$ on $\bigS(\mathfrak{g} [-1])$ that is a derivation of the Schouten bracket.
\end{dfn}

\begin{remark}
Definition \ref{dfn:pinftylie} may be equivalently formulated as follows: A homotopy Lie bialgebra is a graded Lie algebra $\mathfrak{g}$ such that $\mathfrak{g}^*[1]$ is equipped with a polynomial homological Poisson vector field $\hat{d}_\phi$. Any polynomial homological vector field on $\mathfrak{g}^*[1]$ is equivalent to an $L_\infty$-algebra structure on $\mathfrak{g}^*$, and the requirement that $\hat{d}_\phi$ be Poisson expresses a compatibility between the Lie algebra structure on $\mathfrak{g}$ and the $L_\infty$ structure on $\mathfrak{g}^*$ associated to $\hat{d}_\phi$.
\end{remark}

Examples of homotopy Lie bialgebras include differential graded Lie algebras (DGLAs) and Lie bialgebras. Indeed, these are the infinitesimal objects corresponding to $Q$-groups and Poisson Lie groups, respectively.

\section{Homotopy Poisson actions}\label{sec:pinftyact}
Let $\mathcal{M}$ be a homotopy Poisson manifold with Poisson multivector field $\pi$, and let $\mathcal{G}$ be a homotopy Poisson Lie group with Poisson multivector field $\phi$.
\begin{dfn}
A \emph{homotopy Poisson action} of $\mathcal{G}$ on $\mathcal{M}$ is a graded group action such that the action map $\sigma: \mathcal{M} \times \mathcal{G} \to \mathcal{M}$ is a homotopy Poisson morphism.
\end{dfn}
The following proposition gives an infinitesimal description of homotopy Poisson actions, generalizing that of Poisson actions given by Lu and Weinstein \cite{lu-wei}.
\begin{prop}\label{prop:infaction}
 Let $\sigma: \mathcal{M} \times \mathcal{G} \to \mathcal{M}$ be a right homotopy Poisson action, and let $\rho: \mathfrak{g} \to \mathfrak{X}(\mathcal{M})$ be the induced infinitesimal action.  Then the map $\hat{\rho} := \bigS(\rho[-1]) : \bigS(\mathfrak{g} [-1]) \to \vect^\bullet(\mathcal{M})$ is a morphism of differential Gerstenhaber algebras.
\end{prop}
\begin{proof}
That $\hat{\rho}$ respects brackets follows, as in the case of ordinary Lie group actions, from the fact that $\sigma$ is a right action.  It remains to show that $\hat{\rho}$ respects differentials.

The map $\hat{\rho}$ is uniquely characterized by the property that $(0,\linv{v}) \in \vect^\bullet(\mathcal{M} \times \mathcal{G})$ is $\sigma$-related to $\hat{\rho}(v)$ for any $v \in \bigS(\mathfrak{g}[-1])$, where $\linv{v}$ is the left-invariant multivector field associated to $v$.

By the definition of homotopy Poisson action, we have that $(\pi,\phi)$ is $\sigma$-related to $\pi$. Since the Schouten bracket preserves relatedness, we then have that $[(\pi,\phi), (0,\linv{v})]$ is $\sigma$-related to $[\pi, \hat{\rho}(v)]$. On the other hand, $[(\pi,\phi), (0,\linv{v})] = (0, [\phi,\linv{v}]) = (0, d_\phi\!\! \linv{v})$, so we conclude that $\hat{\rho}(\hat{d}_\phi v) = [\pi, \hat{\rho}(v)] = d_\pi \hat{\rho}(v)$.
\end{proof}

\begin{lemma}\label{lemma:quotient}
 Let $\mathcal{M}$ be a homotopy Poisson manifold, and let $\mathcal{G}$ be a homotopy Poisson Lie group with a homotopy Poisson action on $\mathcal{M}$.  Then the algebra $C^\infty(\mathcal{M})^\mathcal{G}$ of $\mathcal{G}$-invariant functions on $\mathcal{M}$ is closed under the multibrackets $\beta_\ell$.  Therefore, if the action is free and proper (so that the quotient $\mathcal{M}/\mathcal{G}$ is a graded manifold and we may identify $C^\infty(\mathcal{M})^\mathcal{G}$ with $C^\infty(\mathcal{M}/\mathcal{G})$), then $\mathcal{M}/\mathcal{G}$ inherits a homotopy Poisson structure.
\end{lemma}
\begin{proof}
Let $f_1, \dots, f_\ell \in C^\infty(\mathcal{M})$ be $\mathcal{G}$-invariant. Equivalently, $\tau^* f_i = \sigma^* f_i$, where $\tau: \mathcal{M} \times \mathcal{G} \to \mathcal{M}$ is the map given by projection onto $\mathcal{M}$. Clearly, $\tau$ is a homotopy Poisson morphism, so, letting $\tilde{\beta}_\ell$ denote the $\ell$-ary bracket on $C^\infty(\mathcal{M} \times \mathcal{G})$, we have
\begin{equation*}
	\begin{split}
		\tau^* \beta_\ell(f_1, \dots, f_\ell) &= \tilde{\beta}_\ell(\tau^*f_1, \dots, \tau^*f_\ell) \\
&= \tilde{\beta}_\ell(\sigma^*f_1, \dots, \sigma^*f_\ell) \\
&= \sigma^* \beta_\ell(f_1, \dots, f_\ell).
	\end{split}
\end{equation*}
Thus, $\beta_\ell(f_1, \dots, f_\ell)$ is $\mathcal{G}$-invariant.
\end{proof}

\section{Hamiltonian actions}\label{sec:hamiltonian}

Let $\mathcal{S}$ be a degree $1$ symplectic $Q$-manifold; that is, $\mathcal{S}$ is a ($\mathbb{Z}$-)graded manifold equipped with a nondegenerate degree $-1$ Poisson structure and a degree $1$ Poisson vector field $Q_\mathcal{S}$ such that $Q_\mathcal{S}^2 = 0$. Suppose that a homotopy Poisson Lie group $(\mathcal{G}, \phi)$ has a Hamiltonian right action on $\mathcal{S}$ with action map $\sigma: \mathcal{S} \times \mathcal{G} \to \mathcal{S}$ and equivariant moment map $\mu: \mathcal{S} \to \mathfrak{g}^*[1]$.

Let $L$ and $R$ be the maps from $T^*[1]\mathcal{G}$ to $\mathfrak{g}^*[1]$ given by left- and right-translation, respectively. Specifically, if $\linv{v}$ and $\rinv{v}$ denote, respectively, the left- and right-invariant multivector fields associated to $v \in \bigS(\mathfrak{g}[-1]) \subseteq C^\infty(\mathfrak{g}^*[1])$, then $L$ and $R$ are given by $R^* v = \rinv{v}$ and $L^* v = \linv{v}$.

We identify $T^*[1]\mathcal{G}$ with $\mathfrak{g}^*[1] \times \mathcal{G}$ via right-translation, i.e.\ by identifying $R$ with the projection map $\mathfrak{g}^*[1] \times \mathcal{G} \to \mathfrak{g}^*[1]$. Using this identification, we may lift the moment map to a map $\tilde{\mu} := (\mu \times 1): \mathcal{S} \times \mathcal{G} \to T^*[1]\mathcal{G}$. The equivariance property of $\mu$ is then equivalent to the commutativity of the diagram
\begin{equation}\label{diag:equivariance}
 \xymatrix{\mathcal{S} \times \mathcal{G} \ar^{\tilde{\mu}}[r] \ar_{\sigma}[d] & T^*[1]\mathcal{G} \ar^{L}[d] \\ \mathcal{S} \ar^{\mu}[r] & \mathfrak{g}^*[1]}.
\end{equation}

We construct a vector field $\Phi$ on $\mathcal{S} \times \mathcal{G}$, given by $\Phi(f) = Q_\mathcal{S} (f)$ for $f \in C^\infty(\mathcal{S})$ and $\Phi(\alpha) = \tilde{\mu}^* d_\phi \alpha$ for $\alpha \in C^\infty(\mathcal{G})$.  On the right side of the latter equation, $\alpha$ is viewed as a fiberwise-constant function on $T^*[1]\mathcal{G}$ (or, equivalently, as a $0$-vector field on $\mathcal{G}$).

\begin{lemma}
 If the moment map $\mu$ is a $Q$-manifold morphism (that is, if $Q_\mathcal{S}$ is $\mu$-related to $\hat{d}_\phi$), then $\Phi^2 = 0$.
\end{lemma}
\begin{proof}
Let $\tau$ denote the projection map from $\mathcal{S} \times \mathcal{G}$ to $\mathcal{S}$.  Observe that the diagram
\begin{equation}
 \xymatrix{\mathcal{S} \times \mathcal{G} \ar^{\tilde{\mu}}[r] \ar_{\tau}[d] & T^*[1]\mathcal{G} \ar^{R}[d] \\ \mathcal{S} \ar^{\mu}[r] & \mathfrak{g}^*[1]}
\end{equation}
presents $\mathcal{S} \times \mathcal{G}$ as the fiber product $\mathcal{S} \bitimes{\mu}{R} T^*[1]\mathcal{G}$. The multiplicativity of $\phi$ implies that $R$ is a $Q$-manifold morphism \cite{me:qgpd}. If $\mu$ is also a $Q$-manifold morphism, then it follows that the product vector field $(Q_\mathcal{S}, d_\phi) \in \mathfrak{X}(\mathcal{S} \times T^*[1]\mathcal{G})$ is tangent to the fiber product. It is immediate from the definition of $\Phi$ that, in this case, the restriction of $(Q_\mathcal{S}, d_\phi)$ to the fiber product is equal to $\Phi$. The equation $\Phi^2=0$ is then a consequence of the fact that $(Q_\mathcal{S}, d_\phi)^2 = (Q_\mathcal{S}^2, d_\phi^2) = 0$.
\end{proof}

Thus, if $\mu$ is a $Q$-manifold morphism, then $\Phi$ gives $\mathcal{S} \times \mathcal{G}$ the structure of a $Q$-manifold.

\begin{dfn}\label{dfn:qham}
The action of $\mathcal{G}$ on $\mathcal{S}$ is called \emph{$Q$-Hamiltonian} if the moment map $\mu$ and the action map $\sigma$ are both $Q$-manifold morphisms.
\end{dfn}

\begin{thm}\label{thm:reduction}
Let $\mathcal{G}$ be a flat homotopy Poisson Lie group with a $Q$-Hamiltonian action on a degree $1$ symplectic $Q$-manifold $\mathcal{S}$. If $0$ is a regular value of $\mu$ and the action of $\mathcal{G}$ on  $\mu^{-1}(0)$ is free, then the homological vector field on $\mathcal{S}$ descends to a symplectic homological vector field on $\mu^{-1}(0)/\mathcal{G}$.
\end{thm}
\begin{proof}
Since $0$ is a regular value, the ideal $\mathcal{I}$ of functions on $\mathcal{S}$ that vanish on $\mu^{-1}(0)$ is generated by $\{ \mu^*v : v \in \mathfrak{g}[-1] \}$. The equivariance property of $\mu$ implies that $\{\mu^*v, \mu^*w\} = \mu^*[v,w]$ for all $v,w \in \mathfrak{g}[-1]$. Therefore $\mathcal{I}$ is closed under the Poisson bracket, which means that $\mu^{-1}(0)$ is a coisotropic submanifold of $\mathcal{S}$.

Let $v \in \mathfrak{g}[-1]$. Then $Q_\mathcal{S} (\mu^* v) = \mu^* (\hat{d}_\phi v)$, since $\mu$ is assumed to be a $Q$-manifold morphism. The flatness of $\phi$ ensures that $\hat{d}_\phi v$ does not have a constant component, so $\hat{d}_\phi v$ vanishes at $0$, and therefore $Q_\mathcal{S} (\mu^* v)$ vanishes on $\mu^{-1}(0)$. This proves that $\mathcal{I}$ is $Q_\mathcal{S}$-invariant.

If $\mathcal{G}$ is connected, then the algebra of functions on the symplectic quotient is $N(\mathcal{I})/\mathcal{I}$, where $N(\mathcal{I})$ is the Poisson normalizer of $\mathcal{I}$. The fact that $\mathcal{I}$ is $Q_\mathcal{S}$-invariant implies that $N(\mathcal{I})$ is $Q_\mathcal{S}$-invariant, and it follows that $Q_\mathcal{S}$ descends to $N(\mathcal{I})/\mathcal{I}$.
The descended vector field inherits the properties of being symplectic and homological from $Q_\mathcal{S}$.

In the case where $\mathcal{G}$ is not connected, we need to furthermore show that the restriction of $Q_\mathcal{S}$ to $\mu^{-1}(0)$ descends to the global quotient. For this, we consider the restriction of the action map $\sigma$ to $\sigma_0 : \mu^{-1}(0) \times \mathcal{G} \to \mu^{-1}(0)$. The restriction of the vector field $\Phi$ to $\mu^{-1}(0) \times \mathcal{G}$ is simply the product vector field $(Q_\mathcal{S}, \phi_1)$, where $\phi_1 \in \mathfrak{X}(\mathcal{G})$ is the vector field component of $\phi$. The $Q$-Hamiltonian property of the action implies that $\sigma_0$ is a morphism of $Q$-manifolds. Thus, we may apply Lemma \ref{lemma:quotient} to the action of $(\mathcal{G}, \phi_1)$ on $(\mu^{-1}(0), Q_\mathcal{S})$ to conclude that $Q_\mathcal{S}$ descends to the quotient.
\end{proof}

\begin{remark}
	In the proof of Theorem \ref{thm:reduction}, the condition that $\sigma$ be a $Q$-manifold morphism was not needed in the case where $\mathcal{G}$ is connected. The reason for this is that, when $\mathcal{G}$ is connected, it can be shown that the condition that $\mu$ be a $Q$-manifold morphism implies the same for $\sigma$.
\end{remark}

\section{Homotopy Poisson actions revisited}\label{sec:pinftyact2}

In ordinary symplectic geometry, a standard example of Hamiltonian action is that of cotangent lift. Specifically, if $G$ is a Lie group with a free and proper action on a manifold $M$, then there is a natural way to lift the action of $G$ on $M$ to a Hamiltonian action of $G$ on $T^*M$, where the symplectic quotient may be identified with $T^*(M/G)$. In a sense, this allows us to view ordinary quotients as examples of symplectic quotients.

In this section, we interpret quotients in the homotopy Poisson category in terms of symplectic quotients, via a supergeometric version of the cotangent lift construction. Throughout this section, let $(\mathcal{M}, \pi)$ be a homotopy Poisson manifold, and let $(\mathcal{G}, \phi)$ be a flat homotopy Poisson Lie group with a free and proper right homotopy Poisson action $\sigma: \mathcal{M} \times \mathcal{G} \to \mathcal{M}$.

First, we will describe the shifted cotangent lift action of $\mathcal{G}$ on $T^*[1]\mathcal{M}$.

Consider the pullback bundle $\sigma^*(T^*[1]\mathcal{M})$. We may make the identification $C^\infty(\sigma^*(T^*[1]\mathcal{M})) = C^\infty(\mathcal{M} \times \mathcal{G}) \otimes_\sigma \vect^\bullet(\mathcal{M})$. There is a natural push-forward map $\sigma_*: \vect^\bullet(\mathcal{M} \times \mathcal{G}) \to C^\infty(\sigma^*(T^*[1]\mathcal{M}))$, satisfying the property that $\sigma_* Z = 1 \otimes X$ whenever $Z \in \vect^\bullet(\mathcal{M} \times \mathcal{G})$ is $\sigma$-related to $X \in \vect^\bullet(\mathcal{M})$. The map $\sigma_*$ has a right inverse $\iota: C^\infty(\sigma^*(T^*[1]\mathcal{M})) \to \vect^\bullet(\mathcal{M} \times \mathcal{G})$, whose image is the space of horizontal multivector fields. Thus, if $Z$ is a horizontal multivector field on $\mathcal{M} \times \mathcal{G}$, then $\iota \circ \sigma_* Z = Z$.

Let $\tilde{\sigma}$ and $\tilde{\tau}$ be maps from $\sigma^*(T^*[1]\mathcal{M})$ to $T^*[1]\mathcal{M}$, given by $\tilde{\sigma}^*X = 1 \otimes X$ and $\tilde{\tau}^*X = \sigma_*(X,0)$ for $X \in \vect^\bullet(M)$. Letting $\tau$ be the projection map from $\mathcal{M} \times \mathcal{G}$ to $\mathcal{G}$, we have that the following diagrams commute:
\begin{align*}
 \xymatrix{\sigma^*(T^*[1]\mathcal{M}) \ar^{\tilde{\tau}}[r] \ar[d] & T^*[1]\mathcal{M} \ar[d] \\ \mathcal{M} \times \mathcal{G} \ar^{\tau}[r] & \mathcal{M}} && \xymatrix{\sigma^*(T^*[1]\mathcal{M}) \ar^{\tilde{\sigma}}[r] \ar[d] & T^*[1]\mathcal{M} \ar[d] \\ \mathcal{M} \times \mathcal{G} \ar^{\sigma}[r] & \mathcal{M}}
\end{align*}
We may identify $\sigma^*(T^*[1]\mathcal{M})$ with $T^*[1]\mathcal{M} \times \mathcal{G}$ by taking $\tilde{\tau}$ to be the projection map, and we may then interpret $\tilde{\sigma}$ as the action map for a right action of $\mathcal{G}$ on $T^*[1]\mathcal{M}$.

\begin{dfn}
 The action of $\mathcal{G}$ on $T^*[1]\mathcal{M}$ associated to the map $\tilde{\sigma}:\sigma^*(T^*[1]\mathcal{M}) = T^*[1]\mathcal{M} \times \mathcal{G} \to T^*[1]\mathcal{M}$ is the \emph{(shifted) cotangent lift} of $\sigma$.
\end{dfn}

We emphasize that this cotangent lift construction is simply a supergeometric analogue of the ordinary cotangent lift construction. In particular, the homotopy Poisson structures on $\mathcal{G}$ and $\mathcal{M}$ do not play any role in the construction.

Next, we will show that $\tilde{\sigma}$ is a Hamiltonian action, where the moment map arises from the infinitesimal action $\mathfrak{g} \to \mathfrak{X}(\mathcal{M})$.

Let $\rho: \mathfrak{g} \to \mathfrak{X}(\mathcal{M})$ be the infinitesimal action associated to $\sigma$, characterized by the property that $(0,\linv{v}) \in \mathfrak{X}(\mathcal{M} \times \mathcal{G})$ is $\sigma$-related to $\rho(v)$ for $v \in \mathfrak{g}$. Similarly, the infinitesimal action associated to $\tilde{\sigma}$ is a map $\tilde{\rho} : \mathfrak{g} \to \mathfrak{X}(T^*[1]\mathcal{M})$, characterized by the property that $\tilde{v}$ is $\tilde{\sigma}$-related to $\tilde{\rho}(v)$ for $v \in \mathfrak{g}$, where $\tilde{v} \in \mathfrak{X}(\sigma^*(T^*[1]\mathcal{M}))$ is determined by the equations
\begin{align}
 \tilde{v}(\tilde{\tau}^*X) &= 0, \label{eqn:tildev1} \\
\tilde{v}(\alpha \otimes 1) &= \linv{v}(\alpha) \otimes 1 \label{eqn:tildev2}
\end{align}
for $X \in \vect^\bullet(\mathcal{M})$ and $\alpha \in C^\infty(\mathcal{G})$.
\begin{lemma}\label{lemma:momentlift}
 $\tilde{\rho}(v) = [\rho(v), \cdot]$ for all $v \in \mathfrak{g}$.
\end{lemma}
\begin{proof}
 First, we claim that
\begin{equation}\label{eqn:tildev3}
 \tilde{v}(\theta) = \sigma_* [(0,\linv{v}), \iota(\theta)]
\end{equation}
 for all $\theta \in C^\infty(\sigma^*(T^*[1]\mathcal{M}))$. Indeed, for $X \in \vect^\bullet(\mathcal{M})$, we have
\begin{equation*}
 \begin{split}
  \sigma_* [(0,\linv{v}), \iota(\tilde{\tau}^*X)] &=  \sigma_* [(0,\linv{v}), \iota \circ \sigma_*(X,0)] \\
&=  \sigma_* [(0,\linv{v}), (X,0)] \\
&= 0,
 \end{split}
\end{equation*}
and for $\alpha \in C^\infty(\mathcal{G})$, we have
\begin{equation*}
 \begin{split}
  \sigma_* [(0,\linv{v}), \iota(\alpha \otimes 1)] &=  \sigma_* [(0,\linv{v}), (0,\alpha)] \\
&=  \sigma_* (0,\linv{v}(\alpha)) \\
&= \linv{v}(\alpha) \otimes 1.
 \end{split}
\end{equation*}
Thus we see that \eqref{eqn:tildev3} is valid, since it agrees with \eqref{eqn:tildev1} and \eqref{eqn:tildev2}.

To prove the lemma, we need to show that $\tilde{v}$ is $\tilde{\sigma}$-related to $[\rho(v), \cdot]$ for $v \in \mathfrak{g}$. Using \eqref{eqn:tildev3} and the fact that $\iota(1 \otimes X)$ is $\sigma$-related to $X$, we have
\begin{equation*}
 \begin{split}
  \tilde{v}(\tilde{\sigma}^*X) &= \tilde{v}(1 \otimes X) \\
&= \sigma_*[(0, \linv{v}), \iota(1 \otimes X)] \\
&= 1 \otimes [\rho(v),X] \\
&= \tilde{\sigma}^*([\rho(v),X])
 \end{split}
\end{equation*}
for $X \in \vect^\bullet(\mathcal{M})$, as desired.
\end{proof}

An immediate consequence of Lemma \ref{lemma:momentlift} is the following:
\begin{cor}
 The cotangent lift action $\tilde{\sigma}$ is Hamiltonian, with moment map $\mu: T^*[1]\mathcal{M} \to \mathfrak{g}^*[1]$ given by $\mu^*(v) = \rho(v)$ for $v \in \mathfrak{g}[-1]$.
\end{cor}

Finally, we will show, using the fact that the action of $\mathcal{G}$ on $\mathcal{M}$ is homotopy Poisson, that the cotangent lift action is $Q$-Hamiltonian, and therefore the hypotheses of Theorem \ref{thm:reduction} are satisfied.

Following the construction of \S\ref{sec:hamiltonian}, we lift the moment map to a map $\tilde{\mu}: \sigma^*(T^*[1]\mathcal{M}) \to T^*[1]\mathcal{G}$, which in this case is given by $\tilde{\mu}^*\theta = \sigma_*(0,\theta)$ for $\theta \in \vect^\bullet(\mathcal{G})$. Then we define the vector field $\Phi \in \mathfrak{X}(\sigma^*(T^*[1]\mathcal{M}))$ by
\begin{align*}
 \Phi(\tilde{\tau}^*X) &= \tilde{\tau}^*(d_\pi X),\\
\Phi(\alpha \otimes 1) &= \tilde{\mu}^* d_\phi \alpha,
\end{align*}
for $X \in \vect^\bullet(\mathcal{M})$ and $\alpha \in C^\infty(\mathcal{G})$.

\begin{lemma}\label{lemma:philift}
$\Phi = \sigma_* \circ d_{(\pi,\phi)} \circ \iota$.
\end{lemma}
\begin{proof}
It suffices to check that $\Phi$ and $\sigma_* \circ d_{(\pi,\phi)} \circ \iota$ agree on functions of the form $\tilde{\tau}^*X$ and $\alpha \otimes 1$ for $X \in \vect^\bullet(\mathcal{M})$ and $\alpha \in C^\infty(\mathcal{G})$. Thus we compute
\begin{equation*}
 \begin{split}
   \Phi(\tilde{\tau}^*X) &= \sigma_*(d_\pi X,0)\\
&= \sigma_* \circ d_{(\pi,\phi)}(X,0) \\
&= \sigma_* \circ d_{(\pi,\phi)} \circ \iota \circ \sigma_*(X,0) \\
&= \sigma_* \circ d_{(\pi,\phi)} \circ \iota \circ \tilde{\tau}^*X
 \end{split}
\end{equation*}
and
\begin{equation*}
 \begin{split}
\Phi(\alpha \otimes 1) &= \sigma_* (0, d_\phi \alpha)\\
&= \sigma_* \circ d_{(\pi,\phi)} \alpha \\
&= \sigma_* \circ d_{(\pi,\phi)} \circ \iota(\alpha \otimes 1).\qedhere
 \end{split}
\end{equation*}
\end{proof}

\begin{prop}\label{prop:quotientham}
The cotangent lift $\tilde{\sigma} : T^*[1]\mathcal{M} \times \mathcal{G} \to T^*[1]\mathcal{M}$ is a $Q$-Hamiltonian action.
\end{prop}
\begin{proof}
Proposition \ref{prop:infaction} implies that the moment map $\mu$ is a $Q$-manifold morphism. It remains to show that $\tilde{\sigma}$ is a $Q$-manifold morphism.

 The fact that $\sigma$ is a homotopy Poisson map means that, if $Z \in \vect^\bullet(\mathcal{M} \times \mathcal{G})$ is $\sigma$-related to $X \in \vect^\bullet(\mathcal{M})$, then $d_{(\pi,\phi)} Z$ is $\sigma$-related to $d_\pi X$. For all $X \in \vect^\bullet(\mathcal{M})$, we have that $\iota(1 \otimes X)$ is $\sigma$-related to $X$. Therefore $d_{(\pi,\phi)} \circ \iota(1 \otimes X)$ is $\sigma$-related to $d_\pi X$, or in other words,
\begin{equation*}
 \sigma_* \circ d_{(\pi,\phi)} \circ \iota(1 \otimes X) = 1 \otimes d_\pi X.
\end{equation*}
Using Lemma \ref{lemma:philift}, we then have $\Phi(1 \otimes X) = 1 \otimes d_\pi X$, which may be rewritten as $\Phi(\tilde{\sigma}^*X) = \tilde{\sigma}^*(d_\pi X)$. Thus we conclude that $\Phi$ is $\tilde{\sigma}$-related to $d_\pi$.
\end{proof}

Proposition \ref{prop:quotientham} allows us to apply Theorem \ref{thm:reduction}, and the following result is a straightforward consequence.

\begin{thm}\label{thm:reductionaction}
 Let $\mathcal{M}$ be a homotopy Poisson manifold, and let $\mathcal{G}$ be a flat homotopy Poisson Lie group with a free and proper homotopy Poisson action on $\mathcal{M}$. Then the homotopy Poisson structure on the quotient $\mathcal{M}/\mathcal{G}$ is given by the induced homological vector field on the symplectic quotient $T^*[1]\mathcal{M} // \mathcal{G} = T^*[1](\mathcal{M}/\mathcal{G})$.
\end{thm}

\begin{example}
 Let $M$ be a Poisson manifold, and let $G$ be a Poisson Lie group with a free and proper Poisson action on $M$. Then Theorem \ref{thm:reductionaction} says that the reduced symplectic $NQ$-manifold $T^*[1]M//G$ corresponds to the quotient Poisson manifold $M/G$.
Thus we have an interpretation of Poisson quotients in terms of reduction of degree $1$ symplectic $NQ$-manifolds.
\end{example}

\section{Homotopy Poisson structures of degree $n$}\label{sec:higher}

The framework of ``reduction by homotopy Poisson Lie group action'' described in \S\ref{sec:hamiltonian} is specific to degree $1$ symplectic $Q$-manifolds. In order to deal with symplectic $Q$-manifolds of higher degree, we need to consider analogues of homotopy Poisson Lie groups in higher degrees. In this section, we briefly give definitions and state results for homotopy Poisson structures of degree $n$. We omit proofs, since those for the case $n=1$ generalize in a straightforward way.

A \emph{homotopy Poisson algebra structure of degree $n$} on $\mathcal{A}$ is defined exactly as in Definition \ref{dfn:palg}
except that $\tilde{\mathcal{A}}$  is now defined to be the degree $n$ suspension $\mathcal{A}[n]$, so that $\tilde{\mathcal{A}}_i = \mathcal{A}_{i+n}$. If $\beta_\ell$ is thought of as a bracket on $\mathcal{A}$, then $\beta_\ell$ is of degree $1+(1-\ell)n$.

A \emph{homotopy Poisson manifold of degree $n$} is a graded manifold $\mathcal{M}$ whose algebra of functions $C^\infty(\mathcal{M})$ is equipped with a degree $n$ homotopy Poisson algebra structure of finite type. A degree $n$ homotopy Poisson structure on a graded manifold $\mathcal{M}$ may be described by a degree $n+1$ element $\pi \in C^\infty(T^*[n]\mathcal{M})$ such that $\{\pi,\pi\}=0$. Here, the bracket is the canonical degree $-n$ Poisson bracket on $T^*[n]\mathcal{M}$. The map $\pi \mapsto d_\pi$, where $d_\pi = \{\pi,\cdot\}$ gives a bijection between degree $n$ homotopy Poisson structures and polynomial symplectic homological vector fields on $T^*[n]\mathcal{M}$.

A \emph{homotopy Poisson Lie group of degree $n$} is a graded Lie group equipped with a homotopy Poisson manifold structure such that the multiplication map is a homotopy Poisson morphism. A \emph{homotopy Lie bialgebra of degree $n$} is a graded Lie algebra $\mathfrak{g}$ equipped with a differential on $\bigS(\mathfrak{g}[-n])$ that is a derivation of the degree $-n$ Schouten bracket. Such a differential on $\bigS(\mathfrak{g}[-n])$ may equivalently be viewed as a polynomial homological Poisson vector field on $\mathfrak{g}^*[n]$, or an $L_\infty$-algebra structure on $\mathfrak{g}^*[n-1]$ that is compatible with the Lie algebra structure on $\mathfrak{g}$.

We remark that a $Q$-group (resp.\ DGLA) may be viewed as a homotopy Poisson Lie group (resp.\ homotopy Lie bialgebra) of any degree.

Let $\mathcal{S}$ be a degree $n$ symplectic $Q$-manifold, and let $\mathcal{G}$ be a flat degree $n$ homotopy Poisson Lie group with a Hamiltonian action on $\mathcal{S}$. The moment map $\mu: \mathcal{S} \to \mathfrak{g}^*[n]$ may be lifted to a map $\tilde{\mu}: \mathcal{S} \times \mathcal{G} \to T^*[n]\mathcal{G}$, which may then be used to define a vector field $\Phi$ on $\mathcal{S} \times \mathcal{G}$, as in \S\ref{sec:hamiltonian}. As in Definition \ref{dfn:qham}, the action is called $Q$-Hamiltonian if the moment map and action map are both $Q$-manifold morphisms. Then Theorem \ref{thm:reduction} generalizes as follows.
\begin{thm}\label{thm:reductionhigher}
Let $\mathcal{G}$ be a flat degree $n$ homotopy Poisson Lie group with a $Q$-Hamiltonian action on a degree $n$ symplectic $Q$-manifold $\mathcal{S}$. If $0$ is a regular value of $\mu$ and the action of $\mathcal{G}$ on $\mu^{-1}(0)$ is free, then the homological vector field on $\mathcal{S}$ descends to a symplectic homological vector field on $\mu^{-1}(0)/\mathcal{G}$.
\end{thm}

We conclude by describing two examples of degree $2$ homotopy Lie bialgebras.

\begin{example}[Left-central Courant algebras]
A \emph{Courant algebra} \cite{bcg} over a Lie algebra $\mathfrak{g}$ is a vector space $\mathfrak{a}$ equipped with a bilinear bracket $\llbracket \cdot, \cdot \rrbracket$ and a map $p: \mathfrak{a} \to \mathfrak{g}$ such that, for all $a_1, a_2, a_3 \in \mathfrak{a}$,
\begin{enumerate}
\item $\llbracket a_1, \llbracket a_2, a_3 \rrbracket \rrbracket = \llbracket \llbracket a_1,  a_2\rrbracket, a_3  \rrbracket + \llbracket a_2, \llbracket a_1, a_3 \rrbracket \rrbracket$,
\item $p( \llbracket a_1, a_2 \rrbracket) = [p(a_1), p(a_2)]$.
\end{enumerate}
A Courant algebra $\mathfrak{a}$ is \emph{left-central} if \ $\llbracket h, a \rrbracket = 0$ for all $h \in \ker p$, $a \in \mathfrak{a}$.

Let $\mathfrak{a}$ be a left-central Courant algebra, and let $\mathfrak{h} := \ker p$. We may form a graded vector space $\tilde{\mathfrak{g}} := \mathfrak{h}[2] \oplus \mathfrak{a}[1] \oplus \mathfrak{g}$, where elements of $\mathfrak{h}$ and $\mathfrak{a}$ are viewed as being of degree $-2$ and $-1$, respectively. The bracket on $\mathfrak{a}$ induces a graded Lie bracket $\{\cdot, \cdot\}$ on $\tilde{\mathfrak{g}}$, defined as
\begin{equation}\label{eqn:ecabracket}\begin{split}
&\{(h_1,a_1,g_1, h_2,a_2,g_2)\} = \\
&= \left( \llbracket b_1, h_2 \rrbracket + \llbracket a_1, a_2 \rrbracket + \llbracket a_2, a_1 \rrbracket - \llbracket b_2, h_1 \rrbracket, \llbracket b_1, a_2 \rrbracket - \llbracket b_2, a_1 \rrbracket, [g_1,g_2] \right),
\end{split}
\end{equation}
where $b_1, b_2 \in \mathfrak{a}$ are such that $p(b_i) = g_i$.  The left-central property of $\mathfrak{a}$ implies that (\ref{eqn:ecabracket}) is well-defined.

The exact sequence
\begin{equation*}
\xymatrix{0 \ar[r] & \mathfrak{h} \ar[r] & \mathfrak{a} \ar[r] & \mathfrak{g} \ar[r] & 0}
\end{equation*}
determines a differential on $\tilde{\mathfrak{g}}$. This differential is a derivation of the Lie bracket, so $\tilde{\mathfrak{g}}$ is a DGLA.

Let $\mathcal{S}$ be the degree $2$ symplectic $NQ$-manifold corresponding to a Courant algebroid $E$. Then an infinitesimal $Q$-Hamiltonian action of $\tilde{\mathfrak{g}}$ on $\mathcal{S}$ may be shown to be equivalent to what Bursztyn, Cavalcanti, and Gualtieri \cite{bcg} called an ``extended action with moment map'' of $\mathfrak{a}$ on $E$. In this case, their notion of moment map reduction coincides with $Q$-Hamiltonian reduction \cite{bcmz}.
\end{example}

\begin{example}[Matched pairs]
 Let $\tilde{\mathfrak{g}} = \mathfrak{h} [1] \oplus \mathfrak{g}$ be a graded Lie algebra; in other words, $\mathfrak{g}$ is a Lie algebra equipped with an action on the vector space $\mathfrak{h}$. The action can be dualized to an action of $\mathfrak{g}$ on $\mathfrak{h}^*$.

A quadratic degree $2$ homotopy Lie bialgebra structure on $\tilde{\mathfrak{g}}$ is given by a graded Lie algebra structure on $\tilde{\mathfrak{g}}^*[1] = \mathfrak{h}^* \oplus \mathfrak{g}^*[1]$, which is equivalent to a Lie algebra structure on $\mathfrak{h}^*$ and an action of $\mathfrak{h}^*$ on $\mathfrak{g}^*$. This action may be dualized to an action of $\mathfrak{h}^*$ on $\mathfrak{g}$.

There is a compatibility condition between the graded Lie algebra structures on $\tilde{\mathfrak{g}}$ and $\tilde{\mathfrak{g}}^*[1]$. Expressed in terms of the actions of $\mathfrak{g}$ and $\mathfrak{h}^*$ on each other, the compatibility condition is exactly that of a \emph{matched pair} \cite{majid:matched1,majid:matched2} of Lie algebras. Thus, Lie algebra matched pairs form a special case of degree $2$ homotopy Lie bialgebras.
\end{example}

\bibliographystyle{abbrv}


\section{Introduction}
Face recognition is one of the most commonly used techniques in biometrics authentication applications, of access and video surveillance control, this is due to its advantageous features. In a face recognition system, the individual is subject to a varied contrast and brightness lighting, background. This three-dimensional shape, when it is part of a two-dimensional surface, as is the case of an image, can lead to significant variations. The human face is an object of three-dimensional nature. This object may be subject to various rotations not only flat but also space and also subject to deformations due to facial expressions. The shape and characteristics of this object also change over time.\\

In these last years, a number of methods have been proposed for the recognition of human faces. In spite of the results obtained in this domain, the automatic face recognition stays one of very difficult problem. Several methods were developed for 2D face recognition. However, it has a certain number of limitations related to the orientation of the face or laying, lighting, facial expression. These last years, we talk more and more about 3D face recognition technology as solution of 2D face recognition problems. There are methods of 3D face recognition based on the use of three-dimensional information of the human face in the 3D space. Existing approaches that address the problem of 3D face recognition can be classified into several categories of approaches: Geometric or Local approaches 3D, Bronstein et al [1, 2] propose a new representation based on the isometric nature of the facial surface, Samir et al [3, 4] use 2D and 3D facial curves for analyzing the facial surface; Holistic approaches, Heseltine et al [5] have developed two approaches to applying the representations PCA Three-dimensional face, Cook et al [6] present a robust method for facial expression based on Log-Gabor models from images of deep and some approaches are based on face  segmentation can be found in [7 , 8, 9, 10, 11, 12].\\

In this work we present an automatic 3D face recognition system based on facial surface analysis using a Riemannian geometry. For this we take the following steps:\\

- 	Detection of 3D face where the nose end is a reference point.\\

-	Iso-geodesic curves extraction using a 3D Fast Marching algorithm.\\

-	Compute of geodetic distance between two iso-geodesic curves using mathematical formulas in Riemannian metric.\\

The rest of this paper is organized as follows: Section 2 describes the methodology of the proposed method with its stages: reference point detection, geodesic distance computing, facial curves extraction. Section 3 includes the simulation results and analysis. Section 4 draws the conclusion of this work and possible points for future work.\\

\section{METHODOLOGY}
Given an image of 3D face Shape REtrieval Contest 2008 database (SHRED2008) database our goal is to realize an automatic 3D face recognition system based on the computation of the geodesic distance between the reference point (nose tip) and the other points in the 3D face surface. So, our algorithm is divided to four main steps, first: Reference Point Detection, in this paper we have detected the point of reference (nose tip) manually. Second: Geodesic Distance computation, an effective method to compute a geodesic distance between two points of facial surface is using the Fast Marching as a numerical algorithm for solving the Eikonal equation. Third: facial curves extraction and compute a geodesic distance between each pair of curves. Finally: classification algorithms, in this step we use three types of classification algorithms: the Neural Networks (NN), k-Nearest Neighbor (KNN) and Support Vector Machines (SVM). Figure (1) illustrates the steps of proposed method in our 3D face recognition system.\\
\begin{figure}[H]
\centering
\includegraphics[width=4in]{figure1.pdf}
\caption{Methodology Architecture}
\label{fig_sim}
\end{figure}

\subsection{Reference Point Detection}
The reference point (nose tip) is detected manually or automatically. There are several automatic approaches: L. Ballihi et al are developed an automatic algorithm nose end detection of a 3D face in [13]. This algorithm is based on two cuts of the facial surface. The first is at transverse face of the mass center. The second cut is based on the minimum depth point of the horizontal curve obtained by the first cut. The output of the last cut is a vertical curve and the minimum depth of this curve is the end of 3D face nose. In [14] S. Bahanbin et al use Gabor filters to automatically detect the nose tip. Another method has been used by C. Xu et al in 2004 [15], this method computes the effective energy of each neighbor pixel, then be determined the mean and variance of each neighbor pixel and uses the SVM to specify point end of the nose. L.H. Anuar et al [16] use a local geometry curvature and point signature to detect a nose tip region in 3D face model.\\

In this work we have detected the point of reference $p_{0}$ (nose tip) manually. The following figure (figure2) summarizes the steps to detect the nose tip of a 3D face an image of the database SHREC2008:
\begin{figure}[H]
\centering
\includegraphics[width=3in]{figure2.pdf}
\caption{3D face nose end detection steps: (a) 3D face image; (b) Manual nose tip selection; (c) Reference point detection}
\label{fig_sim}
\end{figure}

\subsection{Geodesic Distance}
The geodesic distance between two points’ p0 and p of 3D face surface is the shortest path between the two points while remaining on the facial surface. In the context of calculating the geodesic distance R. Kimmel and J.A. Sethian [17] propose the method of Fast Marching as a solution of the Eikonal equation.\\

The Eikonal equation given as:
\begin{equation}
   \mid \bigtriangledown_{u} (x) \mid = F(x); \quad x \in \Omega
\end{equation}
with:\\ - $\Omega$ is an open set in $R^{n}$ housebroken limit.\\ - $\bigtriangledown$ denotes the
gradient.\\ - $\mid . \mid$ is the Euclidean norm.\\

The Fast Marching method is a numerical method for solving boundary value problems of the Eikonal equation [17, 18, 19]. The algorithm is similar to the Dijkstra's algorithm [20] and uses that information flows only to the outside from the planting area.\\

We consider a 3D face surface discretized using a triangular mesh with N vertices. Figure (3) shows a 3D face image of the Shape REtrieval Contest 2008 (SHREC2008) database whose obverse surface is discretized into a triangular mesh.
\begin{figure}[H]
\centering
\includegraphics[width=3in]{figure3.pdf}
\caption{3D facial surface discretized on triangular mesh of N vertices}
\label{fig_sim}
\end{figure}
The geodesic distance between two points on a surface is calculated as the length of the shortest path connecting the two points. Using the Fast Marching algorithm on the triangulated surface 3D face, we can compute the geodesic distance between the reference point P0 and the other point’s p constructing the facial surface.\\

The geodesic distance $\delta_{p_{0},p}$ between $p_0$ and p is approximated by the following expression:
\begin{equation}
   \delta_{p_{0},p} = min \gamma (\beta(p_{0},p))
\end{equation}
with:\\ - $\beta(p_{0},p)$ is the path between $p_0$ and according to the facial surface $S$ of the 2D face.\\ - $\gamma (\beta(p_{0},p))$ is the path length.\\

The following figure (Figure4) shows the steps for determining the geodesic distance using a 3D face image of SHREC2008 database.
\begin{figure}[H]
\centering
\includegraphics[width=3in]{figure4.pdf}
\caption{3D face geodesic distance computes Steps: (a) 3D face image; (b) Reference point detection; (c) Discretization by triangular mesh; (d) Geodesic distance computing}
\label{fig_sim}
\end{figure}

\subsection{Facial Curves}
This 3D face recognition method is based on the analysis of facial surfaces by analyzing of facial curves using Riemannian geometry. To extract the curves of a 3D face surface, the first step is to define the real-valued function on this surface [17]. According to the extraction strategy, different types of facial curves can be found: 1- Iso-depth curves, these curves are obtained by the intersection of the 3D face surface with parallel planes perpendicular to the direction of watching. The depth curves located at equal values of z [17]. 2-Iso-radius curves, these curves are determined by the intersection of the facial surfaces with spheres as centre is the reference point of 3D face (nose tip) and variable radius [13]. 3- Iso-geodesic curves are defined as the locations of all points on the facial surface having the same geodesic distance to the reference point chosen (in our case the end of the nose). The geodesic distance between two points on a surface is the shortest path between these two points along of surface [13, 17].\\

In this work, we represent the 3D human face surface by a collection of iso-geodesic curves. To extract the iso-geodesic curves we use the Fast Marching algorithm as a solution of Eikonal equation [19].  Figure (5) present the steps of extracting of iso-geodesic curves in some 3D face images of SHREC2008 database.
\begin{figure}[H]
\centering
\includegraphics[width=3in]{figure5.pdf}
\caption{Extracting facial curves steps: (a) Pre-treated face image, (b) Detection of the reference point, (c) Compute of geodesic distance, (d) Facial curves extraction}
\label{fig_sim}
\end{figure}
Given two points on a face surface S (reference point $p_0$ and $p$), A geodesic distance between $p_0$ and $p$ is defined as the arc length of the shortest path between these two points along the surface and denoted by a Geodesic Distance Function (GDF), which is a continuous function on the facial surface.

\begin{equation}
   F(p_{0},p) = k; \quad k \in [0, + \propto[ and (p_{0},p) \in S
\end{equation}

We can therefore define the facial curves by:
\begin{equation}
   c_k = \{ p \in S \setminus F(p_{0},p) = k \} \subset S; \quad k \in [0, + \propto[
\end{equation}

The function F defines the geodesic distance between $p_0$ and $p$, or the length of the shortest path between these two points while remaining on the surface $S$.\\

This definition allows us to cite three cases $c_k$ according to the values of $k$:\\
\begin{itemize}\setlength{\itemsep}{4mm}
\item If  $k=0$ then  $c_k$  tends towards the reference point $p_0$ : $c_k = \{ p_0 \}$.
\item If  $k\rightarrow\propto$ then  $c_k$  is empty : $c_k=\{ 0\}$.
\item If  $0<k<\propto$ then  $c_k$  approaches  $S$ : $c_k =\{p\in S \backslash F(p,p_0) = k \}$.
\end{itemize}
\vspace{0.5cm}
\subsection{Riemannian analysis of facial curves}

To analyze the facial surfaces, we simply analyze the iso-geodesic curves that characterize these 3D face surfaces and compute a geodesic distance between them on a manifold depends on the Riemannian metric. To analyze the curve shape, we use the parameterization by the mathematical function SRVF (Square Root Velocity Function) [30, 31].\\

Let a parameterized closed curve be denoted as $\beta: I \rightarrow R^3$, for a unit interval $I \equiv [0, 2\pi]$, $\beta$ is represented by its $SRVF: q: I \rightarrow R^3$ defined as follow:

\begin{equation}\label{notre}
  q(s)= \frac{\beta'(s)}{(\parallel\beta(s)\parallel)^{\frac{1}{2}}}= \frac{\frac{d\beta(s)}{ds}}{\sqrt{\parallel\frac{d\beta(s)}{ds}\parallel}} \in R^3
\end{equation}
Where,
\begin{itemize}\setlength{\itemsep}{4mm}
\item $s \in I \equiv [0, 2\pi]$.
\item $\|.\|$ is the standard Euclidean norm in $R^3$.
\item $\|q(s)\|$  is the square-root of the instantaneous speed on the curve $\beta$.
\item $\frac{q(s)}{\|q(s)\|}$  is the instantaneous direction at the point $s \in [0, 2\pi]$ along the curve.
\end{itemize}
\vspace{0.5cm}
Thus, the curve $\beta$ can be recovered within a translation, using:
\begin{equation}\label{notre}
  \beta(s)= \int_{0}^{s} q(t)\|q(t)\| \, \mathrm{d}t
\end{equation}

We define the set of closed curves $\beta$ in $R^3$ by:
\begin{equation}\label{notre}
  C=\{q : S^1 \rightarrow R^3  \mid \int_{S^1}q(t)\parallel q(t)\parallel dt=0 \} \subset L(S^1,R^3)
\end{equation}
Where,
\begin{itemize}\setlength{\itemsep}{4mm}
\item $L(S^1,R^3)$. denotes the set of all functions integral $S^1$ to $R^3$
\item $\int_{S^1}q(t)\parallel q(t)\parallel dt$ denotes the total displacement in $R^3$ while moving from the origin of the curve until the end. When $\int_{S^1}q(t)\parallel q(t)\parallel dt=0$ the curve is closed.
\end{itemize}
\vspace{0.5cm}
All 3D closed curves are defined as nonlinear variety in the Helbert space. To analyze the shapes of the iso-geodesic curves and compute a geodesic distances between them, it is important to understand all vectors of their tangent spaces and impose a Riemannian structure. We equip the space of the closed curves of a Riemannian structure using the inner product defined as follows:
\begin{equation}\label{notre}
  <f,g>= \int_{0}^{1} (f(s),g(s)) ds
\end{equation}

Here, $f$ and $g$ are two vectors in the tangent space $T_v(c)$. We can also define $T_v(c)$:

\begin{equation}\label{notre}
 T_v(C)= \{ f:S^1 \rightarrow R^3 \mid <f(s),h(s)>=0, \quad  h \in N_v(c) \}
\end{equation}
$N_v(c)$ is a space of the normal vectors to the face curve.

After a mathematical representation of iso-geodesic curves using Riemannian metric, this metric should invariant certain transformation (translation, rotation, scale) [30]. The question to ask is how to compute the geodesic distance between two closed curves?. To answer this question, we used the approach introduced by Klassen in 2007 [31]. This method use path straightening flows to find a geodesic between two shapes.\\

To compare two facial surfaces, we just compare a pairs of closed curves of these two facial surfaces. Lets $c_1$ and $c_2$ two facial curves (iso-geodesic curves), $q_1$ and $q_2$ are respectively there Square Root Velocity Function ($SRVF$). The geodesic distance between $c_1$ and $c_2$ is computed by:

\begin{equation}\label{notre}
 d(q_1,q_2)= \int_0^1 \sqrt{<\varepsilon\prime(t),\varepsilon\prime(t)>}  dt
\end{equation}

With $\varepsilon$ is a geodesic path determined by the training method, this method is to connect the two curves by an arbitrary path $\alpha$ then update the path repeatedly in the negative direction of the gradient of the energy given by:

\begin{equation}\label{notre}
 E[\alpha] = \frac{1}{2} \int_0^1 <\frac{d}{ds}\alpha(t),\frac{d}{ds}\alpha(t)> dt
\end{equation}

$\varepsilon$ has been shown that the critical points of the energy equation $E[\alpha]$ are geodesic paths in $S$ [31, 13].\\

The facial surfaces $S_1$ and $S_2$ are represented by their iso-geodesic curves collection respectively $\{c_k^1; k \in [0,k_0]\}$ and $\{c_k^2; k \in [0,k_0]\}$, $k$ is a geodesic distance between $p_0$ (reference point) and $p$ two points of facial surface $S$.  The vectors of geodesic distances computed between a pairs of facial curves are used as input vectors of classification algorithms of our automatic facial recognition system.\\

To realize our 3D face recognition system, we use three classification algorithms are: the Neural Networks (NN), k-Nearest Neighbor (KNN) and Support Vector Machines (SVM).

\section{Simulation Results}

In this section we make a series of simulation to evaluate the effectiveness of the proposed approach. These results were performed based on SHREC 2008 database. This database contains total of 427 scans of 61 subjects (45 males and 16 females), for each of these 61 subjects 7 different scans, namely two “frontal”, one “look-up”, one “look-down”, one “smile”, one “laugh” and one “random expression” [21, 22].\\

In this paper, the features were extracted using Iso-Geodesic Curves (I-GC). This method was based on two principal steps: Iso-Geodesic Curves extraction using Fast Marching algorithm as solution of Eiconal equation and compute the length of the geodesic path between each facial curve and its corresponding curve using a Riemannian framework.\\

The following figure (6) summarizes the first experimental results of our simulation:

\begin{figure}[H]
\centering
\includegraphics[width=3.4in]{figure6.pdf}
\caption{Recognition rate in terms of number of facial curves used to represent the human face of SHREC 2008 database for each classification algorithm (NN, KNN and SVM)}
\label{fig_sim}
\end{figure}

Figure 5 shows the recognition rate in terms of number of facial curves used to represent a 3D human faces used in our systems. This figure shows that the images of the SHREC 2008 database are represented using five facial curves.\\

Given a candidate 3D face image ($Img$) of SHREC 2008 database. $Img$ is represented using five iso-geodesic curves. The shortest path between two 3D face images id defined as the sum of the distance between all pairs of corresponding facial curves in the two face images. The feature vector is then formed by the geodesic distances computed on all the curves and its dimension is equal to the number of used iso-geodesic curves (five curves for SHREC 2008 database). These vectors are used as input of classification algorithms of our 3D face recognition systems.

\begin{figure}[H]
\centering
\includegraphics[width=3in]{figure7.pdf}
\caption{Recognition Rate for SHREC 2008 images using three classification algorithms (NN, KNN and SVM)}
\label{fig_sim}
\end{figure}

Figure 7 shows the recognition rate for SHREC 2008 images using three classification algorithms: Neural Networks (NN), K-Nearest Neighbor (KNN) and Support Vector Machines (SVM). The best recognition rate was obtained using Support Vector Machines (SVM) as classification algorithm with recognition rate equal $98,9\%$.\\

In conclusion of this series of results, a summary table (Table2) compares the performance of our face authentication with respect to the performance obtained in other work systems.
\begin{table}[!h]
\centering
\begin{tabular}{|l|p{1.3cm}|p{2.7cm}|c|p{1.3cm}|}
\hline
\bf Date & \bf Reference & \bf Method & \bf Database & \bf Reported performance\\
\hline \hline
2004 & Haar et al [26] & facial Contour Curves & SHREC’08 & 91.1 \%  \\
\hline
2007 & Feng et al [24] & Euclidean Integral Invariants Signature (Closed 3D Curves) & FRGCv2 & 95,0\% \\
\hline
2007 & Samir et al [23] & Planar Curves Levels & Notre Dame & 90,4 \% \\
\hline
2007 & Samir et al [23] & Planar Curves Levels & FSU & 92,0\% \\
\hline
2008 & Daoudi et al [25] & Elastic Deformation Of Facial Surfaces (open paths) & FSU & 92,0 \% \\
\hline
2015 & Ahdid et al [29] & GD+LDA+SVM & SHREC’08 & 93,2\% \\
\hline
2015 & Ahdid et al [29] & GD+PCA+SVM & SHREC’08 & 95,3 \% \\
\hline
\bf 2016 & \bf Our approach & \bf Iso-Geodesic Curves + SVM & \bf SHREC’08 & \bf 98,9\% \\
\hline
\end{tabular}
\caption{COMPARISON OF OUR METHOD WITH OTHER METHODS OBTAINED IN OTHER WORK SYSTEMS}
\end{table}

We can notice that the performance of our automatic 3D face recognition system, In addition our system is perfect in all assessment. Our goal was to improve 3D faces recognition system we affirm based on the results that our goal is achieved.

\section{Conclusion}
In this work, we have presented a 3D face recognition system based on the computation of the geodesic distance between the reference point and the other points in the 3D face surface. This method represents a face surface as a collection of Iso-Geodesic curves and computes a geodesic distance between each pairs of these facial curves. For the classifying step we have implemented algorithms as Neural Networks (NN), K-Nearest Neighbor (KNN) and Support Vector Machines (SVM). Simulation results show us a better recognition rate ($98.9\%$) using SVM classification algorithm.