Super Hamiltonian operators and Lie superalgebras

AbstractA superversion of the formal calculus of variations as developed by I.M. Gel'fand et al. is presented. This superversion is essentially contained in the one given by B.A. Kupershmidt. However, the form presented here is more manageable and suitable for applications. Even as well as odd super Hamiltonian operators are considered. It is proved that with every linear super Hamiltonian operator one can associate a Lie superalgebra structure on the space of (reduced) 1-forms and vice versa. Also a theorem is proved about the connection between 2-cocycles on this Lie superalgebra and super Hamiltonian operators. Finally an application to the sKdV equation is given