Examples of Lie and Balinsky-Novikov algebras related to Hamiltonian operators

Abstract

Abstract We study algebraic properties of Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Special attention is paid to the Poisson brackets of the Lie-Poisson type, related with the special Lie-structures on the differential-topological torus and brane algebras, generalizing those studied before by Novikov-Balinsky and Gelfand-Dorfman. Illustrative examples of Lie and Balinsky-Novikov algebras are discussed in detail. The non-associative structures (induced by derivation and endomorphism) of commutative algebras related to Lie and Balinsky-Novikov algebras are described in depth.