Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators

• Main Authors: Orest D. Artemovych, Alexander A. Balinsky, Denis Blackmore, Anatolij K. Prykarpatski
• Format: Article
• Language: English
• Published: MDPI AG 2018-11-01
• Series: Symmetry
• Subjects: nonassociative algebra; loop algebra; Lie–Poisson structure; Hamiltonian operator; R-structure; toroidal loop algebra; Poisson structure; Hamiltonian system; derivation; Balinsky– Novikov algebra; weak Balinsky–Novikov algebra; weakly deformed Balinsky–Novikov algebra; reduced pre-Lie algebra; fermionic Balinsky–Novikov algebra; Lie algebra; Lie derivation; Leibniz algebra; Riemann algebra
• Online Access: https://www.mdpi.com/2073-8994/10/11/601

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie⁻Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky⁻Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler⁻Kostant⁻Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky⁻Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky⁻Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky⁻Novikov algebras, including their fermionic version and related multiplicative and Lie structures.