Integrable dynamical systems with quadratic and cubic nonlinearities related to graded Lie algebras

1991 ◽  
Vol 32 (7) ◽  
pp. 1720-1732 ◽  
Author(s):  
Zhuhan Jiang ◽  
S. Rauch‐Wojciechowski
Keyword(s):  
Dynamical Systems ◽  
Lie Algebras ◽  
Graded Lie Algebras ◽  
Integrable Dynamical Systems ◽  
Quadratic And Cubic Nonlinearities

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Cohomology of certain $ \mathbb N$-graded Lie algebras

2004 ◽  
Vol 59 (6) ◽  
pp. 1210-1211
Author(s):  
D V Millionshchikov ◽  
A Fialowski
Keyword(s):  
Lie Algebras ◽  
Graded Lie Algebras

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

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 601
Author(s):  
Orest Artemovych ◽  
Alexander Balinsky ◽  
Denis Blackmore ◽  
Anatolij Prykarpatski
Keyword(s):  
Dynamical Systems ◽  
Lie Algebras ◽  
Algebraic Structures ◽  
Loop Algebras ◽  
Type Symmetry ◽  
Novikov Algebras ◽  
Algebra Theory ◽  
Hamiltonian Operators ◽  
Adjoint Space ◽  
Algebraic Scheme

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.

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