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\def\beqar{\begin{eqnarray}}
\def\eeqar{\end{eqnarray}}
\def\Ga{\Gamma}
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\C{{\cal C}}
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\def\NT{{\bf N}_{T}}
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\def\GNT{{^{G}\bf N}_{T}}
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\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 i<j\leq p}(-1)^{b(i,j)}[X_i,X_j]\wedge X_1\wedge
\stackrel{\stackrel{i}{\surd}}{\dots \ldots} 
\stackrel{\stackrel{j}{\surd}}{\ldots \ldots} \wedge X_{p+1}
\inner\varphi \label{BF}
\eeqar
where 
\beqar
& & a(i)=1+i+p(X_i)\sum_{h=1}^{i-1}p(X_h)~, \nonumber \\
& & b(i,j)=i+j+p(X_i)\sum_{h=1}^{i-1}p(X_h)
+p(X_j)\sum_{h=1\atop h\neq i}^{j-1}p(X_h)~. \label{BG}
\eeqar
>From 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}
ENDBODY
