TY - JOUR

T1 - Reduced pre-lie algebraic structures, the weak and weakly deformed Balinsky-Novikov type symmetry algebras and related Hamiltonian operators

AU - Artemovych, Orest D.

AU - Balinsky, Alexander A.

AU - Blackmore, Denis

AU - Prykarpatski, Anatolij K.

N1 - Funding Information:
Acknowledgments: The authors cordially thank Maciej Błaszak, Jan Cies´linski, Antoni Sym and Anatolij Samoilenko for their cooperation and useful discussions during the International Conference in Functional Analysis dedicated to the 125th anniversary of Stefan Banach held on 18–23 September 2017 in Lviv, Ukraine. A.P. thanks also the Department of Mathematical Sciences of the New Jersey Institute of Technology, (Newark, NJ, USA) for the invitation to visit the New Jersey Institute of Technology, during the summer of 2017, where an essential part of this paper was formulated. A.P. thanks the Department of Physics, Mathematics and Computer Science of the Cracov University of Technology for a local research grant F-2/370/2018/DS.

PY - 2018/11/6

Y1 - 2018/11/6

N2 - 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.

AB - 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.

KW - Balinsky- Novikov algebra

KW - Derivation

KW - Fermionic Balinsky-Novikov algebra

KW - Hamiltonian operator

KW - Hamiltonian system

KW - Leibniz algebra

KW - Lie algebra

KW - Lie derivation

KW - Lie-Poisson structure

KW - Loop algebra

KW - Nonassociative algebra

KW - Poisson structure

KW - R-structure

KW - Reduced pre-Lie algebra

KW - Riemann algebra

KW - Toroidal loop algebra

KW - Weak Balinsky-Novikov algebra

KW - Weakly deformed Balinsky-Novikov algebra

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U2 - 10.3390/sym10110601

DO - 10.3390/sym10110601

M3 - Article

AN - SCOPUS:85057885150

VL - 10

JO - Symmetry

JF - Symmetry

SN - 2073-8994

IS - 11

M1 - 601

ER -