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

被引:5
作者
Artemovych, Orest D. [1 ]
Balinsky, Alexander A. [2 ]
Blackmore, Denis [3 ]
Prykarpatski, Anatolij K. [4 ]
机构
[1] Cracow Univ Technol, Inst Math, Ul Warszawska 24, PL-31155 Krakow, Poland
[2] Cardiff Univ, Math Inst, Cardiff CF24 4AG, S Glam, Wales
[3] New Jersey Inst Technol, Dept Math Sci, Newark, NJ 07102 USA
[4] Cracow Univ Technol, Dept Phys Math & Comp Sci, Ul Warszawska 24, PL-31155 Krakow, Poland
来源
SYMMETRY-BASEL | 2018年 / 10卷 / 11期
关键词
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; AUTOMORPHISM STRUCTURE; HEAVENLY EQUATION; AFFINE STRUCTURES; ROOTED TREES; HIERARCHY; SYSTEMS; CLASSIFICATION;
D O I
10.3390/sym10110601
中图分类号
O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];
学科分类号
07 ; 0710 ; 09 ;
摘要
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.
引用
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页数:28
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