Variational Calculus of Supervariables and Related Algebraic Structures

We establish a formal variational calculus of supervariables, which is a combination of the bosonic theory of Gelʹfand–Dikii and the fermionic theory in our earlier work. Certain interesting new algebraic structures are found in connection with Hamiltonian superoperators in terms of our theory. In particular, we find connections between Hamiltonian superoperators and Novikov–Poisson algebras that we introduced in our earlier work in order to establish a tensor theory of Novikov algebras. Furthermore, we prove that an odd linear Hamiltonian superoperator in our variational calculus induces a Lie superalgebra, which is a natural generalization of the super-Virasoro algebra under certain conditions.