Reduced pre-lie algebraic structures, the weak and weakly deformed Balinsky-Novikov type symmetry algebras and related Hamiltonian operators

Orest D. Artemovych, Alexander A. Balinsky, Denis Blackmore, Anatolij K. Prykarpatski

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4 Scopus citations


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.

Original languageEnglish (US)
Article number601
Issue number11
StatePublished - Nov 6 2018

All Science Journal Classification (ASJC) codes

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • General Mathematics
  • Physics and Astronomy (miscellaneous)


  • Balinsky- Novikov algebra
  • Derivation
  • Fermionic Balinsky-Novikov algebra
  • Hamiltonian operator
  • Hamiltonian system
  • Leibniz algebra
  • Lie algebra
  • Lie derivation
  • Lie-Poisson structure
  • Loop algebra
  • Nonassociative algebra
  • Poisson structure
  • R-structure
  • Reduced pre-Lie algebra
  • Riemann algebra
  • Toroidal loop algebra
  • Weak Balinsky-Novikov algebra
  • Weakly deformed Balinsky-Novikov algebra


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