%%%%%%%%%%%%%%%%%%%%%%%% %% %% THIS IS A LATEX FILE %% %% PLEASE RUN IT TWICE BEFORE YOU PRINT IT %% %% THE OUTPUT CONSISTS OF 17 PAGES %% %%%%%%%%%%%%%%%%%%%%%%%%%%% BODY \documentstyle{article} \parskip=1ex \parindent=2.0em \textwidth=14.5cm \textheight=21.09cm \evensidemargin=0.71cm \oddsidemargin=0.71cm \topmargin=0.71cm \begin{document} \large \baselineskip=18pt %%%%%%%%%%%%%%%%%%%%%% MACROS %%%%%%%%%%%%%%%%% \renewcommand{\thefootnote}{\fnsymbol{footnote}} \def\col#1,#2,#3,#4 {\begin{array}{c}#1\\ #2\\ #3\\#4 \end{array}} \def\proof{\bigskip \noindent {\it Proof.} \ } \def\prop #1.{\bigskip \noindent {\bf Proposition #1.} \par} \def\rest #1,#2{{#1}_{\vert #2}} \def\pd#1,#2{\frac{\partial#1}{\partial#2}} \def\lpd#1,#2{\frac{\stackrel{\rightarrow}{\partial#1}}{\partial#2}} \def\rpd#1,#2{\frac{\stackrel{\leftarrow}{\partial#1}}{\partial#2}} \def\d#1{\buildrel .\over{#1}~} \def\dd#1{\buildrel ..\over{#1~}} \def\fraz#1,#2{\frac{#1}{#2}} \def\wh{\widehat} \def\wt{\widetilde} \def\inner{\underline {\ \ }\kern -.1em \raise.3ex\hbox{$\vert $}\,} \def\Ker{\mbox{Ker}} \def\Hom{\mbox{Hom}} \def\Im{\mbox{Im}} \def\Der{\mbox{Der}} \def\rank{\mbox{rk}} \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\beqar{\begin{eqnarray}} \def\eeqar{\end{eqnarray}} \def\Ga{\Gamma} \def\A{{\cal A}} \def\B{{\cal B}} \def\C{{\cal C}} \def\F{{\cal F}} \def\G{{\cal G}} \def\I{{\cal I}} \def\L{{\cal L}} \def\M{{\cal M}} \def\O{{\cal O}} \def\S{{\cal S}} \def\T{{\cal T}} \def\X{{\cal X}} \def\func{\F (M)} \def\vect{\X (M)} \def\form{\X (M)^*} \def\tens{\T^{1}_{1} (M)} \def\sfunc{\G (S)} \def\svect{\X (S)} \def\sform{\X (S)^*} \def\stens{\T^{1}_{1} (M)} \def\hT{\widehat T} \def\cT{\check T} \def\NT{{\bf N}_{T}} \def\HT{{\bf H}_{T}} \def\GNT{{^{G}\bf N}_{T}} \def\GHT{{^{G}\bf H}_{T}} \def\Cinf{C^\infty} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \thispagestyle{empty} \hfill DSMA-TS QM2 256 \hfill February, 1992 \vspace{.5cm} \begin{center} {\Large \bf REMARKS ON THE COMPLETE INTEGRABILITY \\ OF DYNAMICAL SYSTEMS WITH \\ FERMIONIC VARIABLES} \footnote{Supported in part by the italian Ministero dell' Universit\`a e della Ricerca Scientifica e Tecnologica.} \end{center} \vspace{.5cm} \begin{center} {\Large G. Landi$^{1,4}$, G. Marmo$^{2,4}$ and G. Vilasi$^{3,4}$} \end{center} \bigskip \begin{center} $^{1}$ Dipartimento di Scienze Matematiche - Universit\`a di Trieste,\\ P.le Europa, 1 - I-34100 Trieste --- Italy.\\ \smallskip $^{2}$ Dipartimento di Scienze Fisiche - Universit\`a di Napoli,\\ Mostra d'Oltremare, Pad.19 - I-80125 Napoli --- Italy.\\ \smallskip $^{3}$ Dipartimento di Fisica Teorica e S.M.S.A. - Universit\`a di Salerno,\\ Via S. Allende, I-84081 Baronissi (SA) --- Italy.\\ \smallskip $^{4}$ Istituto Nazionale di Fisica Nucleare - Sezione di Napoli\\ Mostra d'Oltremare, Pad.20 - I-80125 Napoli --- Italy. \end{center} \vspace{1.0cm} {\bf Abstract.} We study the r\^{o}le of $(1,1)$ graded tensor field $T$ in the analysis of complete integrability of dynamical systems with fermionic variables. We find that such a tensor $T$ can be a recursion operator if and only if $T$ is even as a graded map, namely, if and only if $p(T) = 0$. We clarify this fact by constructing an odd tensor for two examples, a supersymmetric Toda chain and a supersymmetric harmonic oscillator. We explicitly show that it cannot be a recursion operator not allowing then to build new constants of the motion out of the first two ones in contrast to what usually happens with ordinary, i.e. non graded systems. \pagebreak \noindent {\bf 1. Introduction.} In recent years there has been a renewed interest in completely integrable Hamiltonian systems, specially in connection with the study of integrable quantum field theory, Yang-Baxter algebras and, more recently, quantum groups. Integrability criteria available both in finite and infinite dimensions have been established by methods directly related to group theory and to familiar procedures in classical mechanics [AM], [Ar], by looking at soliton equations as dynamical systems on (infinite-dimensional) phase manifold [DMSV 1-2], [Vi], [GD], [Ma], [KM]. This last approach was also suggested by the occurrence in the Inverse Scattering Transform of a peculiar operator, the so called recursion operator [La], which naturaly fits in this geometrical setting as a mixed tensor field on the phase manifold. This tensor has to satisfies few requirements, the most important being that its Nijenhuis torsion [FN], [Ni] vanishes. There have been several attempts to analyse integrability of fermionic dynamical systems (see for instance, [Ku], [MR], [DR]) and to extend to such systems [DHR], in algorithimic sense at least, results and techniques used for bosonic dynamics and based on the r\^{o}le of recursion operators. In particular, one would like to define a graded Nijenhuis torsion. In this paper we address this issues. We show that a mixed $(1,1)$ graded tensor field $T$ can act as a recursion operator if and only if $T$ is an even map. There are dynamical systems, like supersymmetric Witten's dynamics [Wi] which allow a bi-Hamiltonian description with an even and odd Hamiltonian function and in term of an even and an odd Poisson structure respectively (so that the dynamical vector field is always even) [VPST], [So]. This allows to construct an odd tensor field which could be a good candidate as a recursion operator. We explicitely show that this is not possible. The paper is organized as follows. First, we fix notation and recall the formulation of complete integrability in term of a $(1,1)$ tensor field available in the bosonic case. After a r\'{e}sum\'{e} of graded differential calculus and graded Poisson structures, we analyse a supersymmetric harmonic oscillator and a supersymmetric Toda chain (both are examples of Witten's supersymmetric dynamics). We then prove that for an odd $(1,1)$ tensor, a $(2,1)$ tensor correspondig to its torsion (graded-Nijenhuis torsion) cannot be defined and that a graded $(1,1)$ tensor cannot be a recursion operator unless it is even. Finally, we present some conclusions. \bigskip \bigskip \noindent {\bf 2. Complete Integrability and recursion operators in the bosonic case.} Complete integrability of Hamiltonian systems with finitely many degrees of freedom is exhaustively characterized by the Liouville-Arnold theorem [AM], [Ar]. Here we briefly recall an alternative characterization in term of an invariant (under the dynamical evolution) (1,1) tensor field $T$. Example of such tensors can be constructed also for systems with infinitely many degrees of freedom so that the approach described could be of use in the latter cases as well. We shall deal only with smooth, i.e. $\Cinf$ objects, and notations will follows as close as possible those of [AM] and [MSSV]. In particular if $M$ is a (finite dimensional) ordinary manifold we denote by $\func$ the ring of real valued functions on $M$, by $\vect$ the Lie algebra of vector fields, by $\form$ its dual of forms and by $\tens$ the mixed $(1,1)$ tensor fields. Associated with every $T \in \tens$ there are two endomorphisms of $\vect$ and $\form$ which are defined by \beqar & & \hT : \vect \longrightarrow \vect ~, ~~~ \cT : \form \longrightarrow \form ~, \nonumber \\ & & T (X, \alpha) =: \hT X \inner \alpha =: X \inner \cT \alpha~,~~~~ \forall ~X \in \vect ~~,~~\alpha \in \form~. \label{AA} \eeqar The {\it Nijenhuis tensor} (or {\it torsion}) of $T$ is the $(2,1)$ tensor field $\NT$ defined by [FN], [Ni] \beq \NT(X,Y; \alpha) =: \HT(X,Y) \inner \alpha ~, \label{AB} \eeq \noindent where $\HT : \vect \times \vect \rightarrow \vect $ is the $\func$-linear map given by \beq \HT(X,Y) =: \hT^{2} [X,Y] + [\hT X,\hT Y] - \hT [\hT X,Y] - \hT [X,\hT Y]~. \label{AC} \eeq Equivalently, this can be written as \beq \HT(X,Y) = [ \wh{L_{\hT X} T} - \hT \circ \wh{L_{X} T} ](Y) ~~,~~\forall ~X~,~Y \in \vect~. \label{AD} \eeq To simplify our notation, in the following, when no confusion arises, we shall denote both the endomorphisms $\hT$ and $\cT$ with the same symbol, namely $T$. >From the very definition (\ref{AB}) it is clear that the vanishing of the tensor $\NT$ is equivalent to the vanishing of $\HT$, namely $\NT \equiv 0$ iff $\HT \equiv 0$. The following proposition has been proved in [DMSV 2] \prop 1. A dynamical vector field $\Ga$ which admits a mixed tensor field $T$, which is invariant ($L_{\Ga} T = 0$), with vanishing Nijenhuis torsion, diagonalizable with doubly degenerate eigenvalues $\lambda$, without stationary points $(d\lambda \neq 0)$ is separable integrable and Hamiltonian, i.e. is a separable completely integrable Hamiltonian system. \bigskip The proof is given observing that: $\NT = 0$ implies the integrability, in the Frobenius sense, of the eigenspaces of T; $L_{\Ga} T = 0$ implies the separability of $\Ga$ along the eigenmanifolds, in dynamics with 1-degree of freedom, each of which having a constant of motion. \bigskip A $(1,1)$ tensor field with the previously stated properties, acts as a `recursion operator' [Ma], [GD], i.e. when iteratively applied to $\Ga$ one produces symmetries $\Ga_k = \hT^k \Ga$ or constants of motion $H_k$ by $dH_k = \cT^k dH$. \bigskip The main property of the tensor field $T$ in the analysis of complete integrability of its infinitesimal automorphisms is the vanishing of its Nijenhuis tensor $\NT = 0$. It is then, plausible that a suitable generalization of such a condition could play an important r\^{o}le in analysing the integrability of dynamical systems with fermionic degrees of freedom. Moreover, it seems natural to think that such a generalization could come from a graded generalization of some of the following relations which are available in the bosonic case : \begin{description} \item[a.] $\NT = 0 \Longrightarrow ImT$ is a Lie algebra. \item[b.] $\NT = 0$ , $d(TdH) = 0 \Longrightarrow d(T^k dH) = 0$~. \item[c.] $\NT = 0 \Longleftrightarrow d_T \circ d_T = 0$~; here $d_T$ is a suitable generalization of the exterior derivative associated with any (1,1) tensor field [MFLMR]. \item[d.] $T =: \Lambda_1^{-1} \circ \Lambda_2$~, $\NT = 0 \Longleftrightarrow \Lambda_1 + \Lambda_2$ satisfies the Jacobi identity. Here $\Lambda_1$ and $\Lambda_2$ are two Poisson structures. \item[e.] $\omega (X,Y) =: [TX,Y] + [X,TY] - T[X,Y]$~; $T \omega(X,Y) = [TX,TY]$ (this is the same as $\NT = 0$) $\Longleftrightarrow [X,Y]_{\lambda} =: [X,Y] + \lambda \omega(X,Y)$ satisfies the Jacobi identity for any value of the real parameter $\lambda$. \end{description} \noindent One could expect that some, if not all, of the previous relations do not hold true in the graded situation. Before we proceed with the analysis of the graded Nijenhuis condition we shall give a brief review of the graded differential calculus on supermanifolds which will be followed by the study of some simple examples. \bigskip \bigskip \noindent {\bf 3. Graded differential calculus.} We review some fundamentals of supermanifold theory [DW],[Rog] while refering to the literature for a mathematically coherent definition [Rot], [BB]. In the following, by graded we shall always mean ${\bf Z}_2$-graded. The basic algebraic object is a real exterior algebra $B_L=(B_L)_0\oplus(B_L)_1$ with $L$ generators. An $(m,n)$ dimensional supermanifold is a topological manifold $S$ modelled over the ``vector superspace'' \beq B_L^{m,n}=(B_L)_0^m\times(B_L)_1^n \label{BA} \eeq by means of an atlas whose transition functions fulfil a suitable ``supersmoothness'' condition. A supersmooth function $f:U\subset B_L^{m,n}\to B_L$ has the usual superfield expansion \beq f(x^1\dots x^m,\theta^1\dots \theta^n)= f_0(x)+\sum_{\alpha=1}^nf_\alpha(x)\,\theta^\alpha +\dots+f_{1\dots n}(x)\,\theta^1\dots \theta^n \label{BB} \eeq where the $x$'s are the even (Grassmann) coordinates, the $\theta$'s are the odd ones, and the dependence of the coefficient functions $f_{\dots}(x)$ on the even variables is fixed by their values for real arguments. We shall denote by $\sfunc$ and $\G (U)$ the graded ring of supersmooth $B_L$-valued functions on $S$ and $U \subset S$ respectively. The class of supermanifolds which, up to now, turns out to be relevant for applications in physics is given by the De Witt supermanifolds. They are defined in terms of a coarse topology on $B_L^{m,n}$, called the De Witt topology, whose open sets are the counterimages of open sets in ${\bf R}^m$ through the body map $\sigma^{m,n}: B_L^{m,n} \to {\bf R}^m$. An $(m,n)$ supermanifold is De Witt if it has an atlas such that the images of the coordinate maps are open in the De Witt topology. A De Witt $(m,n)$ supermanifold is a locally trivial fibre bundle over an ordinary $m$-manifold $S_0$ (called the body of $S$) with a vector fibre [Rog]. This make not a surprise the fact that, modulo some technicalities, a De Witt supermanifold can be identified with a Berezin-Konstant supermanifold [Be], [Ko]. \smallskip The graded tangent space $TS$ is constructed in the following manner. For each $x\in S$, let $\G (x)$ be the germs of functions at $x$ and denote by $T_xS$ the space of graded $B_L$-linear maps $X : \G (x) \to B_L$ which satisfy Leibnitz rule. Then, $T_xS$ is a free graded $B_L$-module of dimension $(m,n)$, and the disjoint union $\bigcup_{x\in S}T_xS$ can be given the structure of a rank $(m,n)$ super vector bundle over $S$, denoted by $TS$. The sections $\svect$ of $TS$ are a graded $\sfunc$-module and are identified with the graded Lie algebra $Der\sfunc$ of derivations of $\sfunc$. Derivations (or vector fields) are said to be even (or odd) if they are even (or odd) as maps (satisfying in addition a graded Leibnitz rule) from $\sfunc \to \sfunc$. A local basis is given by \beq \pd{},{x^1}~, \dots , \pd{},{x^m}~, \pd{},{\theta^1}~, \dots , \pd{},{\theta^n}~. \label{BD} \eeq \bigskip \noindent {\bf Remark.} Unless explicitely stated, by using a partial derivative we shall always mean a left derivative, namely a derivative acting from left. In general, if $z^i = (x^j, \theta^k)$, when acting on any homogeneous function $f \in \sfunc$, left and right derivative are related by \beq \lpd{},{z^i} f = (-1)^{p(z^i)[p(f)+1]} f \rpd{},{z^i}~,~~~ i \in \{1, \dots, m+n \}~. \label{BE} \eeq \bigskip In a similar way one defines the cotangent space and bundle. $T_x^{\star}S$ is the space of graded $B_L$-linear maps from $ T_{x}(S) \to B_L$ and $T^{\star}S = \bigcup_{x\in S}T_{x}^{\star}S$. $T_x^{\star}S$ is a free graded $B_L$-module of dimension $(m,n)$, while $T^{\star}S$ is a rank $(m,n)$ super vector bundle over $S$. The sections $\sform$ of $T^{\star}S$ are a graded $\sfunc$-module and are identified with the graded $\sfunc$-linear maps from $Der\sfunc \to \sfunc$. They are the 1-forms on $S$. Forms are said to be even (or odd) if they are even (or odd) as maps $\vect \to \sfunc$. In general, a $p$ covariant and $q$ contravariant graded tensor is any graded $\sfunc$-multilinear map $\alpha :\svect \times \cdots \times \svect \times \sform \times \cdots \times \sform \longrightarrow \sfunc$ ($p~~\svect$ factors and $q~~\sform $ factors). The collection of all rank $(p,q)$ tensors is a graded $\sfunc $-module. A graded {\it p-form} is a skew-symmetric covariant graded tensors of rank $p$. We denote by $\Omega^{p}(S)$ the collection of all p-forms. The {\it exterior differential} on $S$ is defined by letting $X\inner df=X(f)\ \forall f\in\sfunc,\,X\in \svect$ and is extended to maps $\Omega^p(S)\to\Omega^{p+1}(S),\ p\geq 0$, in the usual way, so that $d^2=0$. If $X_i\in \svect$ are homogeneous elements, \beqar & & X_1\wedge\dots\wedge X_{p+1}\inner d\varphi =: \sum_{i=1}^{p+1}(-1)^{a(i)}\, X_i(X_1\wedge \stackrel{\stackrel{i}{\surd}}{\ldots \ldots} \wedge X_{p+1}\inner\varphi) \nonumber \\ & &+\sum_{1\leq iFrom definition one has that $p(d) = 0$. The Lie derivative $L_{(\cdot)}$ of forms is defined by \beqar && L_{(\cdot)} : \svect \times \Omega^p(S) \to \Omega^p(S)~, \nonumber \\ && L_{X} = X \inner \circ d + d \circ X\inner~,~~\forall X \in \svect~. \label{BH} \eeqar Clearly, $p(L_{X}) = p(X)$. The Lie derivative of any tensor product can be defined in an obvious manner by requiring the Leibnitz rule and can be extended to any tensor by using linearity. \bigskip Suppose now that we have a tensor $T \in \stens$ which is homogeneous of degree $p(T)$. Again we can define two graded endomorphisms of $\svect$ and $\sform$ by the formul{\ae} (in the following two formul{\ae} $X, Y$ are homogeneous elements in $\svect$ while $\alpha$ is any element in $\sform$) \beqar & & \hT : \svect \longrightarrow \svect ~, ~~~ \cT : \sform \longrightarrow \sform ~, \nonumber \\ & & T (X, \alpha) =: \hT X \inner \alpha =: (-1)^{p(X)p(T)} X \inner \cT \alpha~. \label{BL} \eeqar We could be tempted to define a graded Nijenhuis torsion of $T$ by a relation analogous to (\ref{AB}) \beqar \GNT(X,Y; \alpha) &=:& \GHT(X,Y) \inner \alpha ~, \nonumber \\ \GHT(X,Y) &=:& \hT^{2} [X,Y] + (-1)^{p(T)p(X)}[\hT X,\hT Y] - \hT [\hT X,Y] \nonumber \\ &&~~ - (-1)^{p(T)p(X)} \hT [X,\hT Y]~. \label{BM} \eeqar \prop 2. The map $\GHT : \svect \times \svect \rightarrow \svect $ defined in (\ref{BM}) is $\sfunc$-linear and graded antisymmetric if and only if $p(T) = 0$. \proof Just compute. \bigskip \noindent {\bf Remark.} When $p(T)=1$, the map defined in (\ref{BM}) is not antisymmetric nor linear also over even function, also when it is restrict to even vector fields. Therefore eqs. (\ref{BL}) and (\ref{BM}) define a graded tensor (which is in addition graded antisymmetric) if and only if $p(T) = 0$. \bigskip \bigskip \noindent {\bf 4. Poisson supermanifold.} We briefly describe how to introduce super Poisson structures on a $(m,n)$-dimensional supermanifold $S$ [Be], [Le]. For additional results see also [CI]. As before, we shall denote by $z^i = (x^j, \theta^k)~,~~ i\in \{1, \dots, m+n \}$ the local coordinates on $S$. The following proposition is in [Be] and can be proved by direct calculations \newpage \prop 3. Let $\vert \vert \omega^{i j} \vert \vert $ be a $(m+n) \times (m+n)$ matrix (depending upon the point $z \in S$) with the following properties: \begin{description} \item[1.] the elements $\omega^{i j}$ are homogeneous with parity $p(\omega^{i j}) = p(z^i) + p(z^j) + p(\omega)$ and $p(\omega)$ not depending on the indices $i$ and $j$~; \item[2.] \beq \omega^{j i} = - (-1)^{[p(z^i) + p(\omega)][p(z^j) + p(\omega)]} \omega^{i j}~; \label{CA} \eeq \item[3.] \beqar (-1)^{[p(z^i) + p(\omega)][p(z^l) + p(\omega)]} \omega^{i s} \lpd{},{z^s} \omega^{j l} + (-1)^{[p(z^l) + p(\omega)][p(z^j) + p(\omega)]} \omega^{l s} \lpd{},{z^s} \omega^{i j} \nonumber \\ + (-1)^{[p(z^j) + p(\omega)][p(z^i) + p(\omega)]} \omega^{j s} \lpd{},{z^s} \omega^{l i} = 0~. \label{CB} \eeqar \end{description} Then, the following bracket \beq \{ F, G \} =: F \rpd{},{z^i} \omega^{i j} \lpd{},{z^j} G \label{CC} \eeq \noindent makes $\sfunc$ a Lie superalgebras (Poisson superstructure). \bigskip \bigskip We have two different kind of structures according to the fact that $p(\omega) = 0$ (even Poisson structure) or $p(\omega) = 1$ (odd Poisson structure). Indeed, one can check that the bracket (\ref{CC}) has properties \beqar && \{ F, G \} = - (-1)^{[p(F) + p(\omega)][p(G) + p(\omega)]}\{ G, F \}~; \label{CD} \\ && ~~\nonumber \\ && (-1)^{[p(F) + p(\omega)][p(H) + p(\omega)]} \{ \{ F, G \}, H \} + (-1)^{[p(G) + p(\omega)][p(F) + p(\omega)]} \{ \{ G, H \}, F \} \nonumber \\ && ~~~~~~~~~~~ + (-1)^{[p(H) + p(\omega)][p(G) + p(\omega)]} \{ \{ H, F \}, G \} = 0~. \label{CE} \eeqar We infer from (\ref{CD}) and (\ref{CE}) that, when thought of as elements of the Poisson superalgebra, homogeneous elements of $\sfunc$ preserve their parity if $p(\omega) = 0$, while they change it if $p(\omega) = 1$. \bigskip If the matrix $\vert \vert \omega^{i j} \vert \vert$ is regular, then its inverse $\vert \vert \omega_{i j} \vert \vert~,~ \omega_{ij} \omega^{jk} = \delta_i^k~, $ gives the components of a symplectic structure $\omega = \fraz1,2 dz^{i} \wedge dz^{j} \omega_{j i}$ , namely, $\omega$ is closed and nondegenerate with the properties \beqar && p(\omega_{i j}) = p(z^i) + p(z^j) + p(\omega) \nonumber \\ && \omega_{j i} = - (-1)^{p(z^i) p(z^j)} \omega_{i j}~, \label{CF} \eeqar and $\omega$ is homogeneous with parity just equal to $p(\omega)$. There is also a Darboux theorem [Le] \prop 4. Let $(S, \omega)$ be a (m, n)-dimensional symplectic manifold with $\omega$ homogeneous. Then \begin{description} \item[1.] If $p(\omega) = 0$~, then dim $S = (2r,n)$ and there exist local coordinates such that \beq \omega = d q^{i} \wedge d p^{i} + d \xi^{j} \wedge d \xi^{j}~;~~~ ~~~~~~\omega_{i j} = {\scriptstyle % All this junk is to make a smaller matrix \addtolength{\arraycolsep}{-.5\arraycolsep} \renewcommand{\arraystretch}{0.5} \left( \begin{array}{ccc} \scriptstyle 0 & \scriptstyle {\bf I}_r & \scriptstyle 0 \\ \scriptstyle -{\bf I}_r & \scriptstyle 0 & \scriptstyle0 \\ \scriptstyle 0 & \scriptstyle 0 & \scriptstyle {\bf I}_n \end{array} \scriptstyle\right)}~. \label{CG} \eeq \item[2.] If $p(\omega) = 1$~, then dim $S = (m,m)$ and there exist local coordinates such that \beq \omega = d u^{i} \wedge d\xi^{i}~;~~~~~~ \omega_{i j} = {\scriptstyle % All this junk is to make a smaller matrix \addtolength{\arraycolsep}{-.5\arraycolsep} \renewcommand{\arraystretch}{0.5} \left( \begin{array}{cc} \scriptstyle 0 & \scriptstyle {\bf I}_m \\ \scriptstyle -{\bf I}_m & \scriptstyle 0 \end{array} \scriptstyle\right)}~, \label{CH} \eeq \end{description} \bigskip Having a Poisson structure we can deal with Hamilton equations. From (\ref{CC}), if $H$ is the hamiltonian, the corresponding equations are \beq {\d z}^i = \omega^{i j} \lpd{},{z^j} H \label{CI}. \eeq Now we would like to maintain the possibility of explicitly constructing the flow of (\ref{CI}). This requires that the dynamical evolution be an even vector field. In turn this implies that the Poisson structure and the Hamiltonian function should have the same parity so that in particular, with an odd Poisson structure we need an odd Hamiltonian function. \bigskip \bigskip \newpage \noindent {\bf 5. Examples.} Before we analyze the graded Nijenhuis condition we study few examples. \bigskip \noindent {\bf 5.1 Mixed bosonic-fermionic harmonic oscillator.} The mixed bosonic-fermionic harmonic oscilator in $(2,2)$ dimensions is described with coordinates $(q, p, \eta, \xi)$ and has the following equations of motion \beqar && \d{q} = p~, \nonumber \\ && \d{p} = -q~, \nonumber \\ && \d{\eta} = \xi~, \nonumber \\ && \d{\xi} = -\eta~. \label{DA} \eeqar Equations (\ref{DA}) can be given two Hamiltonian descriptions. The Hamiltonians are: the usual even one \beq H = \fraz1,2 (p^2 + q^2) +i\xi \eta~, \label{DB} \eeq and an odd one \beq K = p\xi + q\eta ~, \label{DC} \eeq \noindent while the two Poisson structures are respectively \beq \Lambda_H = \left( \col 0,-1,0,0 \col1,0,0,0 \col 0,0,i,0 \col 0,0,0,i \right)~,~~~~~ \omega_H = \left( \col 0,1,0,0 \col-1,0,0,0 \col 0,0,-i,0 \col 0,0,0,-i \right)~, \label{DD} \eeq and \beq \Lambda_K = \left( \col 0,0,0,-1 \col 0,0,1,0 \col 0,-1,0,0 \col 1,0,0,0 \right)~,~~~~~ \omega_K = \left( \col 0,0,0,1 \col 0,0,-1,0 \col 0,1,0,0 \col -1,0,0,0 \right)~. \label{DE} \eeq \bigskip \noindent We can construct a a mixed invariant tensor field T by \beq T =: \omega_H \circ \Lambda_K = \left( \col 0,0,0,i \col 0,0,-i,0 \col 1,0,0,0 \col 0,1,0,0 \right) \label{DF} \eeq \noindent However, this odd tensor field ($p(T) = 1$) is not a recursion operator. One can easly find that \beqar && T dK = dH~, \nonumber \\ && T dH = -i(dq) \xi + i (dp) \eta -i(d\eta) p +i(d\xi) q~,~~~ d( T dH) \neq 0~. \label{DG} \eeqar If we evaluate the Poisson brackets of the coordinate variables with the two symplectic structure (\ref{DD}) and (\ref{DE}) we find that \beq \{q, p\}_H = 1~,~~~\{p, q\}_H = -1~,~~~ \{\eta, \eta\}_H = i~,~~~\{\xi, \xi\}_H = i~,~~~ \label{DH} \eeq and \beq \{q, \xi\}_K = 1~,~~~\{\xi, q\}_K = -1~,~~~ \{p, \eta\}_K = -1~,~~~\{\eta, p\}_K = 1~,~~~ \label{DI} \eeq the remaining ones being identically zero. We see that the sum $\{\cdot, \cdot \}_+$ of the two structures is itself a Poisson structure with the property \beq \{ F, G \}_{+} = - (-1)^{p(F) p(G)}\{ G, F \}_{+}~, \label{DL} \\ \eeq but it has not definite parity. Moreover $\{\cdot, \cdot \}_+$ is degenerate. \bigskip \bigskip \noindent {\bf 5.2 Witten Dynamics [Wi].} Interesting examples come from supersimmetric dynamics. It has been shown [VPST], [So] that the dynamics of Witten's Hamilton systems [Wi] can be given a bi-Hamiltonian description with an even Poisson bracket and Grassmann even Hamiltonian or with an odd bracket and Grassmann odd Hamiltonians. Instead of considering the general case we shall study a supersymmetric Toda chain with coordinates $(q,p,\eta,\xi)$. The even Hamiltonian is given by \beq H = \fraz1,2 (p^2 + e^q) + \fraz1,2 i \xi \eta e^{\fraz q,2}~. \label{DM} \eeq With the even Poisson structure \beq \Lambda_H = \left( \col 0,-1,0,0 \col1,0,0,0 \col 0,0,i,0 \col 0,0,0,i \right)~,~~~~~ \omega_H = \left( \col 0,1,0,0 \col-1,0,0,0 \col 0,0,-i,0 \col 0,0,0,-i \right)~, \label{DN} \eeq the equations of motion read \beqar && \d{q} = p~, \nonumber \\ && \d{p} = - \fraz1,2 e^q~ - \fraz1,4 i\xi \eta e^{\fraz q,2}, \nonumber \\ && \d{\eta} = \fraz1,2 \xi e^{\fraz q,2}~, \nonumber \\ && \d{\xi} = - \fraz1,2 \eta e^{\fraz q,2}~. \label{DO} \eeqar Then the following functions are constants of the motion \beqar && K = p \xi + e^{\fraz q,2} \eta~, \nonumber \\ && L = p \eta - e^{\fraz q,2} \xi~, \nonumber \\ && F = i \xi \eta~. \label{DP} \eeqar We can use $K$ in (\ref{DP}) (or $L$) as an alternative Hamiltonian. The corresponding symplectic structure can be written as \beqar \omega_K &=& dq \wedge d \xi + dp \wedge dq ( e^{- \fraz {q},2} \eta) + dp \wedge d\eta (-2 e^{- \fraz {q},2}) + df \wedge dH \nonumber \\ &=& d \{ dq (- \xi) + dp (2 e^{- \fraz {q},2} \eta) + f dH \}~, \label{DQ} \eeqar where $f(q,p,\eta,\xi)$ is a function explicitly given by \beqar && f = A \xi + B \eta \nonumber \\ && A(q,p) = {\fraz{1},{p^2 + e^q}} \left( {\fraz{2p},{\sqrt{p^2 + e^q}}} log ( {\fraz{e^{\fraz{q},{2}}},{p + \sqrt{p^2 + e^q}}} ) + {\fraz{2e^{\fraz{q},{2}}},{\sqrt{p^2 + e^q}}} - 2 \right) \nonumber \\ && B(q,p) = {\fraz{1},{p^2 + e^q}} \left( {\fraz{2e^{\fraz{q},{2}}},{\sqrt{p^2 + e^q}}} log ( {\fraz{e^{\fraz{q},2}},{p + \sqrt{p^2 + e^q}}} ) - {\fraz{2p},{\sqrt{p^2 + e^q}}} - 2 p e^{-\fraz{q},{2}} \right)~. \label{DQA} \eeqar \bigskip \noindent If $\Gamma$ is the dynamical vector field of the Toda system, as given by (\ref{DO}), than, the function $f$ is such that ~$i_{\Gamma}df = e^{-\fraz{q},{2}} \eta$~ and this, in turn, assures that ~$i_{\Gamma}\omega_{K} = dK$. \bigskip It take some algebra to check that the $(1,1)$ tensor field \beq T = \omega_K \circ \Lambda_H ~, \label{DR} \eeq is such that \beqar && T dH = dK~, \nonumber \\ && d (T^2 dH) \neq 0~. \label{DS} \eeqar \bigskip \noindent Again, $T$ in (\ref{DR}) is not a recursion operator. \bigskip \bigskip \noindent {\bf 6. Super Nijenhuis torsion.} One of the most relevant consequences deriving from a (not graded) (1-1) tensor field $T$ with vanishing Nijenhuis torsion is the possibility to generate sequences of exact 1-forms according to \prop 5. \beq \NT = 0,~~ d(T dF) = 0 \Longrightarrow d(T^k dF) = 0~. \label{EA} \eeq \proof Let $\alpha$ be any 1-form. By using the expression of the exterior derivative, after some algebra one finds that \beqar {X \wedge Y} \inner d(T^2 \alpha) &=& \{X \wedge TY + TX \wedge Y\} \inner d(T \alpha) - \{TX \wedge TY\} \inner d\alpha \nonumber \\ &&~~- {\HT (X, Y)}\inner \alpha~. \label{EB} \eeqar Where $\HT$ is defined in (\ref{AC}). Assume now that both $\alpha$ and $T \alpha$ are closed. >From (\ref{EB}) we see that $T^2 \alpha$ is closed if and only if $\HT = 0$, namely if and only if the Nijenhuis torsion of $T$ vanishes. \bigskip Let us analize now the graded situation. Suppose $T$ is a graded $(1,1)$ tensor field which is homogeneous of parity $p(T)$. Then, if $\alpha$ is any 1-form, by using the definition (\ref{BF}), after some (graded) algebra, the analogue of (\ref{EB}) reads \beqar {X \wedge Y} \inner d(T^2 \alpha) &=& { \{(-1)^{p(T)p(Y)} X \wedge TY + (-1)^{p(T)[p(X)+p(Y)]} TX \wedge Y\} } \inner d(T \alpha) \nonumber \\ &&~~- (-1)^{p(T)[p(X)+p(T)]} {TX \wedge TY} \inner d\alpha \nonumber \\ &&~~- (-1)^{p(T)}~ \GHT (X,Y) \inner \alpha \nonumber \\ &&~~+ (-1)^{p(T)p(X)} [ 1 - (-1)^{p(T)}] L_{TX} ( {TY}\inner \alpha)~. \label{EC} \eeqar Where $\GHT$ is defined in (\ref{BM}). \bigskip It is clear then, that for an (1,1) odd tensor a (2,1) tensor corresponding to its torsion (super-Nijenhuis torsion) can be defined only when $p(T) = 0$. The same result is attained with the use of the general approach $d_T \circ d_T = 0$ . \bigskip \bigskip \noindent {\bf 7. Conclusions.} Summing up, we have shown that there are examples of dynamical systems whose dynamical vector field $\Gamma$ admits two Hamiltonian descriptions, odd and even respectively, and that the tensor field T , constructed out of the corresponding Poisson structures is not a recursion operator since it cannot generate new integrals of motion after the first two ones. We have also shown that this fact is general and that for a generic graded $(1,1)$ tensor field $T$ a graded Nijenhuis torsion cannot be defined unless $T$ is even. >From the nature of the proof it seems plausible that a similar theorem should hold true also in infinite dimensions. The `no go' theorem we have proved in our paper does not exhausts, obviously, the analysis of complete integrability for graded Hamiltonian systems. Much more attention must be paid, however, in generalizing to the graded case geometrical structures which play a relevent and natural r\^{o}le in the non graded situation. \bigskip \bigskip \noindent {\bf Acknowledgement} We thank U. Bruzzo for useful remarks. We are very grateful to D. Del Santo and P. Omari for their invaluable help in solving a system of partial differential equations which lead to the second symplectic structure for the Toda system. \bigskip \pagebreak \noindent {\bf REFERENCES} \begin{description} \item[AM] R. Abraham, J.E. Marsden, Foundations of Mechanics (Benjamin / Cummings, Reading, MA, 1978). \item[Ar] V.I. Arnold, Les methodes mathematiques de la Mecanique Classique (Mir, Moskow, 1976). \item[BB] C. Bartocci, U. Bruzzo, D. Hern\'{a}ndez-Ruip\'{e}rez, The Geometry of Supermanifolds (Kluwer, Dordrecht, 1991) \item[Be] F. A. Berezin Introduction to superanalysis (Kirillov ed.) (D. Reidel Publishing Company, Dordrecht, 1987). \item[CI] F. Cantrijn, L.A. Ibort, Introduction to Poisson Supermanifolds, Preprint 1989. \item[DHR] A. Das, When-Jui Huang, S. Roy, J. Math. Phys. {\bf 32} (1991) 2733. \item[DMSV 1] S. De Filippo, G. Marmo, M. Salerno, G. Vilasi, On the Phase Manifold Geometry of Integrable Nonlinear Field Theory, Preprint IFUSA, Salerno (1982), unpublished. \item[DMSV 2] S. De Filippo, G.Marmo, M.Salerno, G.Vilasi, Il N. Cimento {\bf 83B} (1984) 97. \item[DR] A. Das, S. Roy, J. Math. Phys. {\bf 31} (1990) 2145. \item[DW] B. De Witt, `Supermanifolds' (London, Cambridge Univ. Press 1984). \item[FN] A. Fr\"{o}licher, A. Nijenhuis, Indag. Math. {\bf 23} (1956) 338. \item[GD] I.M. Gel'fand, I.Ya. Dorfman, Funct. Anal. {\bf 14} (1980) 71. \item[KM] Y. Kosmann-Schwarzbach, F. Magri, Ann. Inst. H. Poincar\'{e} (Physique Th\'{e}orique) {53} (1990) 35. \item[Ko] B. Kostant, Graded manifolds, graded Lie theory and prequantization. In `Differential Geometric Methods in Mathematical Physics', LNM 570 (Springer-Verlag, Berlin, 1977), 177-306. \item[Ku] B. Kupershmidt, Elements of Superintegrable Systems (D. Reidel Publishing Company, Dordrecht, 1987). \item[La] P. D. Lax, Comm. Pure Appl. Math. {\bf 21} (1968) 467; {\bf 28} (1975)141; Siam Rev. {\bf 18} (1976) 351. \item[Le] D. A. Le\u{i}tes, Soviet Math. Dokl. {\bf 18} (1977) 1277. \item[Ma] F. Magri, J. Math. Phys. {\bf 18} (1978) 1156. \item[MFLMR] G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo, C. Rubano, Phys. Rep. {\bf 188} (1990) 147-284. \item[MR] Y. Manin, A.O. Radul, Comm. Math. Phys. {\bf 98} (1985) 65. \item[MSSV] G. Marmo, E.J. Saletan, A. Simoni, B. Vitale, Dynamical Systems (John Wiley, Chichester, 1985). \item[Ni] A. Nijenhuis, Indag. Math. {\bf 49} (1987) 2. \item[Rog] A. Rogers, J. Math. Phys. {\bf 21, } (1980) 1352; Commun. Math. Phys. {\bf 105} (1986) 375. \item[Rot] M. Rothstein, Trans. AMS {\bf 297} (1986) 159. \item[So] V. A. Soroka, Lett. Math. Phys. {\bf 17} (1989) 201. \item[Vi] G. Vilasi, Phys. Lett. {\bf 94B} (1980) 195. \item[VPST] D. V. Volkov, A. I. Pashnev, V. A. Soroka and V.I.Tkach, JETP Lett. {\bf 44} (1986) 70. \item[Wi] E. Witten, Nucl. Phys. {\bf B188} (1981) 513. \end{description} \end{document}