Loop algebras and their relation to the conformal structure of integrable systems

1990 ◽  
Vol 19 (1) ◽  
pp. 35-44
Author(s):  
Sverrir �lafsson
Keyword(s):  
Integrable Systems ◽  
Conformal Structure ◽  
Loop Algebras

DEFORMATIONS OF LOOP ALGEBRAS AND CLASSICAL INTEGRABLE SYSTEMS: FINITE-DIMENSIONAL HAMILTONIAN SYSTEMS

2004 ◽  
Vol 16 (07) ◽  
pp. 823-849 ◽  
Author(s):  
T. SKRYPNYK
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Integrable Systems ◽  
Integrable Hamiltonian Systems ◽  
Infinite Dimensional ◽  
Loop Algebras ◽  
Finite Dimensional ◽  
Quasigraded Lie Algebras ◽  
Classical Integrable

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras and admit Kostant–Adler scheme. Using them we obtain new integrable hamiltonian systems admitting Lax-type representations with the spectral parameter.

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.

Multi-component matrix loop algebras and their applications to integrable systems

2010 ◽  
Author(s):  
Zhu Li ◽  
Wen Xiu Ma ◽  
Xing-biao Hu ◽  
Qingping Liu
Keyword(s):  
Integrable Systems ◽  
Loop Algebras ◽  
Component Matrix

INTEGRABLE SYSTEMS AND DOUBLE-LOOP ALGEBRAS IN STRING THEORY

1991 ◽  
Vol 06 (16) ◽  
pp. 1525-1531 ◽  
Author(s):  
A. MOROZOV
Keyword(s):  
String Theory ◽  
Integrable Systems ◽  
Integrable Models ◽  
Lagrangian Approach ◽  
Double Loop ◽  
Loop Algebras ◽  
Quantum Deformations ◽  
Conformal Models

Entire string theory in the formalism of first quantization cannot be exhausted by the theory of conformal models (CFTs). It is hardly enough to add only 2-dimensional integrable systems. However, examination of these systems may be of use for future guesses and generalizations. The first purpose is to give a unified treatment of all conformal and integrable models. Some steps in this direction are described in the context of Lagrangian approach. The main implication is the need to study 2-loop algebras (like those of fields on [Formula: see text] surfaces) and their quantum deformations.

scholarly journals Integrable Systems in Four Dimensions Associated with Six-Folds in Gr(4, 6)

2018 ◽  
Vol 2019 (21) ◽  
pp. 6585-6613 ◽  
Author(s):  
Boris Doubrov ◽  
Evgeny V Ferapontov ◽  
Boris Kruglikov ◽  
Vladimir S Novikov
Keyword(s):  
Integrable Systems ◽  
Conformal Structure ◽  
Characteristic Variety ◽  
Four Dimensions ◽  
Quadratic Maps ◽  
All Solutions ◽  
Hermitian Geometry ◽  
Linear Sections ◽  
Plücker Embedding ◽  
Monge Ampere

Abstract Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V. A submanifold X ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this construction where X is a six-fold in Gr(4, 6). The corresponding system Σ(X) reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems Σ(X). These naturally fall into two subclasses. • Systems of Monge–Ampère type. The corresponding six-folds X are codimension 2 linear sections of the Plücker embedding Gr(4, 6)$ \hookrightarrow \mathbb{P}^{14}$. • General linearly degenerate systems. The corresponding six-folds X are the images of quadratic maps $\mathbb{P}^{6}\dashrightarrow \ $Gr(4, 6) given by a version of the classical construction of Chasles. We prove that integrability is equivalent to the requirement that the characteristic variety of system Σ(X) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.

On the Heisenberg Supermagnet Model in (2+1)-Dimensions

2017 ◽  
Vol 72 (4) ◽  
pp. 331-337 ◽  
Author(s):  
Zhao-Wen Yan
Keyword(s):  
Integrable System ◽  
Integrable Systems ◽  
Bäcklund Transformations ◽  
Quadratic Constraints ◽  
Backlund Transformations

AbstractThe Heisenberg supermagnet model is an important supersymmetric integrable system in (1+1)-dimensions. We construct two types of the (2+1)-dimensional integrable Heisenberg supermagnet models with the quadratic constraints and investigate the integrability of the systems. In terms of the gage transformation, we derive their gage equivalent counterparts. Furthermore, we also construct new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.

Sign in / Sign up

Export Citation Format

Share Document