Abstract
There are studied algebraic properties of quadratic Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Their relations both with derivations of symmetric tensor algebras and Yang–Baxter structures on the adjacent Lie algebras are demonstrated. Special attention is paid to quadratic Poisson brackets of Lie–Poisson type, examples of Balinsky–Novikov and Leibniz algebras are discussed. The non-associative structures of commutative algebras related with Balinsky–Novikov, Leibniz, Lie, and Zinbiel algebras are studied in detail.
Highlights
Many integrable Hamiltonian systems, discovered during the last decades, were understood [15,16,24,38] owing to the Lie-algebraic properties of their internal hidden symmetry structures
Their importance for constructing integrable multi-component nonlinear Camassa–Holm type dynamical systems on functional manifolds was demonstrated by Strachan and Szablikowski in [47], where there was suggested in part the Lie-algebraic imbedding of the Novikov algebra into the general Lie–Poisson orbits scheme of classification Lax type integrable Hamiltonian systems
We studied the non-associative structures of commutative algebras related with Balinsky–Novikov, Leibniz, Lie and Zinbiel algebras, having diverse important applications both in theory of integrable dynamical systems and to modern problems of communication technology
Summary
Many integrable Hamiltonian systems, discovered during the last decades, were understood [15,16,24,38] owing to the Lie-algebraic properties of their internal hidden symmetry structures. We have devised a simple Lie-algebraic algorithm, allowing to construct new algebraic structures within which the corresponding linear and quadratic Hamiltonian operators, generated by the corresponding Lie–Poisson structure on the co-adjoint orbits, exist and describe the related integrable multicomponent dynamical systems. In these cases an interesting problem of describing the Balinsky–Novikov and Leibniz type algebras, whose multiplicative structures satisfy some additional tensor r -structure type relationships naturally arises. We studied the non-associative structures of commutative algebras related with Balinsky–Novikov, Leibniz, Lie and Zinbiel algebras, having diverse important applications both in theory of integrable dynamical systems and to modern problems of communication technology
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