scholarly journals Generating of Nonisospectral Integrable Hierarchies via the Lie-Algebraic Recursion Scheme

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 621
Author(s):  
Haifeng Wang ◽  
Yufeng Zhang
Keyword(s):  
Lie Algebra ◽  
Efficient Method ◽  
Integrable Hierarchies ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Conserved Quantities ◽  
Recursion Scheme

In the paper, we introduce an efficient method for generating non-isospectral integrable hierarchies, which can be used to derive a great many non-isospectral integrable hierarchies. Based on the scheme, we derive a non-isospectral integrable hierarchy by using Lie algebra and the corresponding loop algebra. It follows that some symmetries of the non-isospectral integrable hierarchy are also studied. Additionally, we also obtain a few conserved quantities of the isospectral integrable hierarchies.

The Zero Curvature Form of Integrable Hierarchies in the ℤ × ℤ-Matrices

2012 ◽  
Vol 19 (02) ◽  
pp. 237-262
Author(s):  
G. F. Helminck ◽  
A. V. Opimakh
Keyword(s):  
Lie Algebra ◽  
Integrable Hierarchies ◽  
Evolution Equations ◽  
Integrable Hierarchy ◽  
Connection Form ◽  
Minimal Realization ◽  
Band Matrices ◽  
Zero Curvature ◽  
Lax Equations ◽  
Finite Band

In this paper it is shown how one can associate to a finite number of commuting directions in the Lie algebra of upper triangular ℤ × ℤ-matrices an integrable hierarchy consisting of a set of evolution equations for perturbations of the basic directions inside the mentioned Lie algebra. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices. The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute. These Lax equations are shown in a purely algebraic way to be equivalent with zero curvature equations for a collection of finite band matrices, that are the components of a formal connection form. One concludes with the linearization of the hierarchies and the notion of wave matrices at zero, which is the algebraic substitute for a basis of the horizontal sections of the formal connection corresponding to this connection form.

Loop algebra and visaoro symmetries of integrable hierarchies of KP type

2001 ◽  
Vol 78 (3-4) ◽  
pp. 233-253 ◽  
Author(s):  
H. Aratyn ◽  
J.F. Gomes ◽  
E. Nissimov ◽  
S. Pacheva
Keyword(s):  
Integrable Hierarchies ◽  
Loop Algebra

scholarly journals INTEGRABLE HIERARCHIES AND CONTACT TERMS IN u-PLANE INTEGRALS OF TOPOLOGICALLY TWISTED SUPERSYMMETRIC GAUGE THEORIES

1999 ◽  
Vol 14 (07) ◽  
pp. 1001-1013 ◽  
Author(s):  
KANEHISA TAKASAKI
Keyword(s):  
Integrable Hierarchies ◽  
Gauge Theories ◽  
Point Of View ◽  
Coupling Constants ◽  
Integrable Hierarchy ◽  
Supersymmetric Gauge Theories ◽  
Tau Function ◽  
Single Time ◽  
Whitham Equations ◽  
Supersymmetric Gauge

The u-plane integrals of topologically twisted N=2 supersymmetric gauge theories generally contain contact terms of nonlocal topological observables. This paper proposes an interpretation of these contact terms from the point of view of integrable hierarchies and their Whitham deformations. This is inspired by Mariño and Moore's remark that the blowup formula of the u-plane integral contains a piece that can be interpreted as a single-time tau function of an integrable hierarchy. This single-time tau function can be extended to a multitime version without spoiling the modular invariance of the blowup formula. The multitime tau function is comprised of a Gaussian factor eQ(t1,t2,…) and a theta function. The time variables tn play the role of physical coupling constants of two-observables In(B) carried by the exceptional divisor B. The coefficients qmn of the Gaussian part are identified to be the contact terms of these two-observables. This identification is further examined in the language of Whitham equations. All relevant quantities are written in the form of derivatives of the prepotential.

scholarly journals A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

2009 ◽  
Vol 2009 ◽  
pp. 1-14
Author(s):  
Do Ngoc Diep
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Vector Bundles ◽  
Sigma Model ◽  
Fock Space ◽  
Loop Algebra ◽  
Target Space ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
Geometric Langlands

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry groupGL. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundles with connection over the space dual to the Lie algebra of the symmetry groupGL. After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry groupGL. Use the electric-magnetic duality to pass to the Langlands dual Lie groupG. Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra=Lie(G). Use the construction of Fock space reprsentations to have representations of such affine loop algebra. And finally, we have the automorphic representations of the corresponding Langlands-dual Lie groupsG.

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.

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 The Classification, Automorphism Group, and Derivation Algebra of the Loop Algebra Related to the Nappi–Witten Lie Algebra

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Xue Chen
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Loop Algebra ◽  
Isomorphism Classes ◽  
Derivation Algebra

Set L ≔ H 4 ⊗ ℂ R , R ≔ ℂ t ± 1 , and S ≔ ℂ t ± 1 / m m ∈ ℤ + . Then, L is called the loop Nappi–Witten Lie algebra. R -isomorphism classes of S / R forms of L are classified. The automorphism group and the derivation algebra of L are also characterized.

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.

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