Super-Derivations and Associated Standard Super-Potentials

Abstractly defined super-derivations on Fermionic systems on a lattice are studied. The existence and uniqueness of the associated standard super-potential are shown for every super-derivation with the subalgebra of all local operators as its domain. The relation between the standard super-potential of a super-derivation and the standard potential for the square of the super-potential (which is shown to be a derivation in the case of finite range super-potentials) is obtained (by use of local super-Hamiltonian for the super-derivation and local Hamiltonian for the square). As a consequence, a necessary and sufficient condition for a super-derivation to be nilpotent is obtained in terms of the corresponding standard super potential. Examples of translation invariant nilpotent super-derivations are given in the case of super-potentials of finite ranges on a one-dimensional lattice. A merit of considering the super-potential associated with a super-derivation is that the former can be used as free parameters for the latter.