scholarly journals Multidimensional integrable systems and deformations of Lie algebra homomorphisms

2007 ◽  
Vol 48 (9) ◽  
pp. 093502 ◽  
Author(s):  
Maciej Dunajski ◽  
James D. E. Grant ◽  
Ian A. B. Strachan
Keyword(s):  
Lie Algebra ◽  
Integrable Systems ◽  
Algebra Homomorphisms

scholarly journals Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 907 ◽  
Author(s):  
Oğul Esen ◽  
Miroslav Grmela ◽  
Hasan Gümral ◽  
Michal Pavelka
Keyword(s):  
Kinetic Theory ◽  
Lie Algebra ◽  
Particle Motion ◽  
Evolution Equations ◽  
Poisson Brackets ◽  
Hamiltonian Approach ◽  
Symmetric Tensors ◽  
Field Equations ◽  
Algebra Homomorphisms

Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establishes a link from particle motion to evolution of the field variables. This pathway is free from Poisson brackets and Hamiltonian functionals. Momentum realizations (sections on T * T * Q ) of (both compressible and incompressible) Euler’s fluid and Vlasov’s plasma are derived. Poisson mappings relating the momentum realizations with the usual field equations are constructed as duals of injective Lie algebra homomorphisms. The geometric pathway is then used to construct the evolution equations for 10-moments kinetic theory. This way the entire Grad hierarchy (including entropic fields) can be constructed in a purely geometric way. This geometric way is an alternative to the usual Hamiltonian approach to mechanics based on Poisson brackets.

scholarly journals On the Stability of -Derivations and Lie -Algebra Homomorphisms on Lie -Algebras: A Fixed Points Method

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Seong Sik Kim ◽  
Ga Ya Kim ◽  
Soo Hwan Kim
Keyword(s):  
Functional Equation ◽  
Fixed Points ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Additive Functional ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
The Stability ◽  
Algebra Homomorphisms ◽  
Stability Results

We investigate new generalized Hyers-Ulam stability results for -derivations and Lie -algebra homomorphisms on Lie -algebras associated with the additive functional equation:

scholarly journals SYMPLECTIC AND ORTHOGONAL LIE ALGEBRA TECHNOLOGY FOR BOSONIC AND FERMIONIC OSCILLATOR MODELS OF INTEGRABLE SYSTEMS

2001 ◽  
Vol 16 (07) ◽  
pp. 1199-1225 ◽  
Author(s):  
A. J. MACFARLANE ◽  
HENDRYK PFEIFFER ◽  
F. WAGNER
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bethe Ansatz ◽  
Integrable Systems ◽  
Fock Spaces ◽  
Efficient Treatment ◽  
Orthogonal Lie Algebra ◽  
Integral Systems ◽  
Quantum Integrable Models ◽  
Invariant Tensors

To provide tools, especially L-operators, for use in studies of rational Yang–Baxter algebras and quantum integrable models when the Lie algebras so (N)(bn, dn) or sp (2n)(cn) are the invariance algebras of their R matrices, this paper develops a presentation of these Lie algebras convenient for the context, and derives many properties of the matrices of their defining representations and of the ad-invariant tensors that enter their multiplication laws. Metaplectic-type representations of sp (2n) and so (N) on bosonic and on fermionic Fock spaces respectively are constructed. Concise general expressions (see (5.2) and (5.5) below) for their L-operators are obtained, and used to derive simple formulas for the T operators of the rational RTT algebra of the associated integral systems, thereby enabling their efficient treatment by means of the algebraic Bethe ansatz.

SPECTRAL RADIUS ANALYSIS OF MATRICES AND THE ASSOCIATED WITH INTEGRABLE SYSTEMS

2009 ◽  
Vol 23 (24) ◽  
pp. 4855-4879 ◽  
Author(s):  
HONWAH TAM ◽  
YUFENG ZHANG
Keyword(s):  
Lie Algebra ◽  
Spectral Radius ◽  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Soliton Equation ◽  
Spectral Matrix ◽  
Akns Hierarchy ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Set Up

An isospectral problem is introduced, a spectral radius of the corresponding spectral matrix is obtained, which enlightens us to set up an isospectral problem whose compatibility condition gives rise to a zero curvature equation in formalism, from which a Lax integrable soliton equation hierarchy with constraints of potential functions is generated along with 5 parameters, whose reduced cases present three integrable systems, i.e., AKNS hierarchy, Levi hierarchy and D-AKNS hierarchy. Enlarging the above Lie algebra into two bigger ones, the two integrable couplings of the hierarchy are derived, one of them has Hamiltonian structure by employing the quadratic-form identity or variational identity. The corresponding integrable couplings of the reduced systems are obtained, respectively. Finally, as comparing study for generating expanding integrable systems, a Lie algebra of antisymmetric matrices and its corresponding loop algebra are constructed, from which a great number of enlarging integrable systems could be generated, especially their Hamiltonian structure could be computed by the trace identity.

CONFORMALLY REDUCED WZNW THEORY, NEW EXTENDED CHIRAL ALGEBRAS AND THEIR ASSOCIATED TODA TYPE INTEGRABLE SYSTEMS

1992 ◽  
Vol 07 (28) ◽  
pp. 7015-7043 ◽  
Author(s):  
BO-YU HOU ◽  
LIU CHAO
Keyword(s):  
Large Class ◽  
Lie Algebra ◽  
Integrable Systems ◽  
Equations Of Motion ◽  
Chiral Algebras ◽  
Block Diagonal

We propose and analyze a large class of conformal reductions Cons [g(H, d)] of WZNW theory based on the integral gradations of the underlying Lie algebra g. The W bases of the associated W algebras W[g(H, d)] are constructed under the generalized Drinfeld-Sokolov gauge which we call O’Raifeartaigh gauge of the constrained Kac-Moody currents, and the equations of motion of the extended Toda type integrable systems corresponding to these W algebras are also derived. As an example, we construct explicitly the W algebra associated with the (pqp) block diagonal decomposition of sl2p+q, namely W[(pqp)2], and discuss some of the properties thereof.

scholarly journals HIDDEN ALGEBRA OF THREE-BODY INTEGRABLE SYSTEMS

1998 ◽  
Vol 13 (18) ◽  
pp. 1473-1483 ◽  
Author(s):  
ALEXANDER TURBINER
Keyword(s):  
Lie Algebra ◽  
Algebraic Structure ◽  
Integrable Systems ◽  
Differential Operators ◽  
Reduction Method ◽  
Hamiltonian Reduction ◽  
Degree Polynomial ◽  
Infinite Dimensional ◽  
Three Body ◽  
Infinite Dimensional Lie Algebra

It is shown that all three-body quantal integrable systems that emerge in the Hamiltonian reduction method possess the same hidden algebraic structure. All of them are given by a second degree polynomial in generators of an infinite-dimensional Lie algebra of differential operators.

scholarly journals ON A CLASS OF INTEGRABLE SYSTEMS CONNECTED WITH GL(N,ℝ)

2004 ◽  
Vol 19 (supp02) ◽  
pp. 205-216 ◽  
Author(s):  
A. GERASIMOV ◽  
S. KHARCHEV ◽  
D. LEBEDEV
Keyword(s):  
Lie Algebra ◽  
Integrable System ◽  
Integrable Systems ◽  
Cotangent Bundle ◽  
Spectral Curve ◽  
The Other ◽  
Quantum Integrable Systems ◽  
New Class ◽  
Integrable Theory

In this paper we define a new class of the quantum integrable systems associated with the quantization of the cotangent bundle T*(GL(N)) to the Lie algebra [Formula: see text]. The construction is based on the Gelfand-Zetlin maximal commuting subalgebra in [Formula: see text]. We discuss the connection with the other known integrable systems based on T*GL(N). The construction of the spectral tower associated with the proposed integrable theory is given. This spectral tower appears as a generalization of the standard spectral curve for an integrable system.

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