scholarly journals Dynamical systems on quantum tori Lie algebras

1993 ◽  
Vol 155 (3) ◽  
pp. 429-448 ◽  
Author(s):  
Jens Hoppe ◽  
Michail Olshanetsky ◽  
Stefan Theisen
Keyword(s):  
Dynamical Systems ◽  
Lie Algebras ◽  
Quantum Tori

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.

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.

scholarly journals Quantum Tori and the Structure of Elliptic Quasi-simple Lie Algebras

1996 ◽  
Vol 135 (2) ◽  
pp. 339-389 ◽  
Author(s):  
Stephen Berman ◽  
Yun Gao ◽  
Yaroslav S. Krylyuk
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras ◽  
Quantum Tori

scholarly journals Invariant Poisson–Nijenhuis structures on Lie groups and classification

2018 ◽  
Vol 15 (04) ◽  
pp. 1850059 ◽  
Author(s):  
Zohreh Ravanpak ◽  
Adel Rezaei-Aghdam ◽  
Ghorbanali Haghighatdoost
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Symmetry Group ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Text Structure ◽  
Baxter Equation ◽  
Text Structures ◽  
Group A

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ([Formula: see text]-[Formula: see text]) on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra [Formula: see text]. We show that [Formula: see text]-[Formula: see text] structures can be used to find compatible solutions of the classical Yang–Baxter equation (CYBE). Conversely, two compatible [Formula: see text]-matrices from which one is invertible determine an [Formula: see text]-[Formula: see text] structure. We classify, up to a natural equivalence, all [Formula: see text]-matrices and all [Formula: see text]-[Formula: see text] structures with invertible [Formula: see text] on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by [Formula: see text]-matrices on a phase space whose symmetry group is Lie group a [Formula: see text], can be specifically determined.

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