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Variational calculus of supervariables and related algebraic structures. (English) Zbl 1012.37048

The author establishes a formal variational calculus of supervariables, which is a combination of the bosonic theory of Gel’fand-Dikii and the fermionic theory in his earlier work [J. Phys. A, Math. Gen. 28, No. 6, 1681-1698 (1995; Zbl 0852.58043)]. In terms of his theory he finds certain interesting new algebraic structures in connection with Hamiltonian superoperators.

In particular, he finds connections between Hamiltonian superoperators and the Novikov-Poisson algebras that he introduced in [J. Algebra 185, No. 3, 905-934 (1996; Zbl 0863.17003)] in order to establish a tensor theory of Novikov algebras. He also proves that an odd linear Hamiltonian superoperator in his variational calculus induces a Lie superalgebra, which is a natural generalization of the super-Virasoro algebra under certain conditions.

37K05Hamiltonian structures, symmetries, variational principles, conservation laws
37K30Relations of infinite-dimensional systems with algebraic structures
17B80Applications of Lie algebras to integrable systems