92-185 Landi G., Marmo G., Vilasi G.

REMARKS ON THE COMPLETE INTEGRABILITY OF DYNAMICAL SYSTEMS WITH FERMIONIC VARIABLES (34K, LaTeX) Nov 24, 92

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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.

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