## Abstract

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 language | English (US) |
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Article number | 601 |

Journal | Symmetry |

Volume | 10 |

Issue number | 11 |

DOIs | |

State | Published - Nov 6 2018 |

## All Science Journal Classification (ASJC) codes

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

## Keywords

- 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