Hamiltonian operators and related differential-algebraic Balinsky-Novikov, Riemann and Leibniz type structures on nonassociative noncommutative algebras

Abstract

We review main differential-algebraic structures \ lying in background of \ analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative  noncommutative algebras. The related Balinsky-Novikov and \ Leibniz type algebraic structures are derived, a new nonassociative "Riemann" algebra is constructed, deeply related with infinite multi-component Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson  structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. \ As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed investigation both of their structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. \ Subject to these important  aspects we stop in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations. We have also added a short Supplement within which we \ revisited \ the classical Poisson manifold approach, closely related to our construction of \ Hamiltonian operators, generated by nonassociative and noncommutative algebras. In particular, \ we presented its natural and simple generalization allowing effectively to describe  a wide class\ of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras.