A subalgebra of Lie algebra A2 and its associated two types of loop algebras, as well as Hamiltonian structures of integrable hierarchy

2009 ◽  
Vol 50 (5) ◽  
pp. 053519 ◽  
Author(s):  
Dong Huan-he
Keyword(s):  
Lie Algebra ◽  
Integrable Hierarchy ◽  
Hamiltonian Structures ◽  
Loop Algebras

scholarly journals Bi-Integrable Couplings of a New Soliton Hierarchy Associated withSO(4)

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Yan Cao ◽  
Liangyun Chen ◽  
Baiying He
Keyword(s):  
Lie Algebra ◽  
Integrable Hierarchy ◽  
Hamiltonian Structures ◽  
Spectral Problems ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Orthogonal Lie Algebra ◽  
Isospectral Problem

Based on the six-dimensional real special orthogonal Lie algebraSO(4), a new Lax integrable hierarchy is obtained by constructing an isospectral problem. Furthermore, we construct bi-integrable couplings for this hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Hamiltonian structures of the obtained bi-integrable couplings are constructed by the variational identity.

scholarly journals Isomorphisms of Twisted Hilbert Loop Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 453-480
Author(s):  
Timothée Marquis ◽  
Karl-Hermann Neeb
Keyword(s):  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Energy ◽  
Algebra Structure ◽  
Affine Lie Algebras ◽  
Subsequent Work ◽  
Highest Weight ◽  
Loop Algebras ◽  
Highest Weight Representations

Abstract The closest infinite-dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e., real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras , also called affinisations of . They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families for some infinite set J. To each of these types corresponds a “minimal ” affinisation of some simple Hilbert-Lie algebra , which we call standard. In this paper, we give for each affinisation g of a simple Hilbert-Lie algebra an explicit isomorphism from g to one of the standard affinisations of . The existence of such an isomorphism could also be derived from the classiffication of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists that is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of g. In subsequent work, this paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of .

TWO FAMILIES GENERALIZATION OF AKNS HIERARCHIES AND THEIR HAMILTONIAN STRUCTURES

2010 ◽  
Vol 24 (08) ◽  
pp. 791-805 ◽  
Author(s):  
YUNHU WANG ◽  
XIANGQIAN LIANG ◽  
HUI WANG
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

By means of the Lie algebra G1, we construct an extended Lie algebra G2. Two different isospectral problems are designed by the two different Lie algebra G1 and G2. With the help of the variational identity and the zero curvature equation, two families generalization of the AKNS hierarchies and their Hamiltonian structures are obtained, respectively.

A NEW LOOP ALGEBRA AND TWO NEW LIOUVILLE INTEGRABLE HIERARCHY

2007 ◽  
Vol 21 (11) ◽  
pp. 663-673 ◽  
Author(s):  
HUAN-HE DONG
Keyword(s):  
Quadratic Form ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Liouville Integrable

A new loop algebra containing four arbitrary constants is presented, and the corresponding computing formula of constant γ in the quadratic-form identity is obtained in this paper, which can be reduced to a computing formula of constant γ in the trace identity. As application, two new Liouville integrable hierarchy and Hamiltonian structures are derived.

scholarly journals Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy and Its Super Hamiltonian Structures

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Sixing Tao ◽  
Tiecheng Xia
Keyword(s):  
Poisson Bracket ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Integrable Couplings ◽  
Classical Integrable

Nonlinear integrable couplings of super Broer-Kaup-Kupershmidt hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained.

HAMILTONIAN STRUCTURES OF TWO INTEGRABLE COUPLINGS OF THE MODIFIED AKNS HIERARCHY

2007 ◽  
Vol 21 (30) ◽  
pp. 2063-2074 ◽  
Author(s):  
YUFENG ZHANG ◽  
Y. C. HON
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Three Dimensional ◽  
Hamiltonian Structures ◽  
Akns Hierarchy ◽  
Higher Dimensional ◽  
Integrable Couplings

The extension of a three-dimensional Lie algebra into two higher-dimensional ones is used to deduce two new integrable couplings of the m-AKNS hierarchy. The Hamiltonian structures of the two integrable couplings are obtained, respectively. Specially, the complex Hamiltonian structure of the second integrable couplings is given.

TWO ENLARGED LOOP ALGEBRAS FOR OBTAINING NEW INTEGRABLE HIERARCHIES

2011 ◽  
Vol 25 (19) ◽  
pp. 2637-2656
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM ◽  
WEI JIANG
Keyword(s):  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Soliton Solutions ◽  
Loop Algebra ◽  
Darboux Transformations ◽  
Soliton Equation ◽  
Hamiltonian Structures ◽  
Soliton Theory ◽  
Loop Algebras ◽  
Soliton Hierarchy

Taking a loop algebra [Formula: see text] we obtain an integrable soliton hierarchy which is similar to the well-known Kaup–Newell (KN) hierarchy, but it is not. We call it a modified KN (mKN) hierarchy. Then two new enlarged loop algebras of the loop algebra [Formula: see text] are established, respectively, which are used to establish isospectral problems. Thus, two various types of integrable soliton-equation hierarchies along with multi-component potential functions are obtained. Their Hamiltonian structures are also obtained by the variational identity. The second hierarchy is integrable couplings of the mKN hierarchy. This paper provides a clue for generating loop algebras, specially, gives an approach for producing new integrable systems. If we obtain a new soliton hierarchy, we could deduce its symmetries, conserved laws, Darboux transformations, soliton solutions and so on. Hence, the way presented in the paper is an important aspect to obtain new integrable systems in soliton theory.

scholarly journals Integrable triples in semisimple Lie algebras

2021 ◽  
Vol 111 (5) ◽  
Author(s):  
Alberto De Sole ◽  
Mamuka Jibladze ◽  
Victor G. Kac ◽  
Daniele Valeri
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Equivalence Class ◽  
Simple Root ◽  
Root Vector ◽  
Integrable Hierarchy ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Kdv Hierarchy ◽  
Hamiltonian Pde

AbstractWe classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in $${\mathfrak {sl}}_2$$ sl 2 corresponds to the KdV hierarchy, and the triple $$(f,0,e_\theta )$$ ( f , 0 , e θ ) , where f is the sum of negative simple root vectors and $$e_\theta $$ e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy.

scholarly journals Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

2021 ◽  
Vol Volume 1 ◽  
Author(s):  
Mats Vermeeren
Keyword(s):  
Differential Equations ◽  
Integrable Hierarchies ◽  
Kepler Problem ◽  
Lagrange Function ◽  
Lagrangian Formulation ◽  
Integrable Hierarchy ◽  
Poisson Brackets ◽  
Exterior Derivative ◽  
Hamiltonian Structures ◽  
Fundamental Object

Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a Lagrange function but a differential $d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for a field to be a solution to the pluri-Lagrangian problem. In this paper we present a procedure to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an integrable hierarchy of PDEs. As a prelude, we review a similar procedure for integrable ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely related to the Poisson brackets between the corresponding Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as the Boussinesq hierarchy.

New Soliton Hierarchies Associated with the Lie Algebra so(3, ℝ) and their BI-Hamiltonian Structures

2015 ◽  
Vol 75 (1) ◽  
pp. 113-133 ◽  
Author(s):  
Shoufeng Shen ◽  
Liya Jiang ◽  
Yongyang Jin ◽  
Wen-Xiu Ma
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structures
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