scholarly journals Integrability structures of the generalized Hunter–Saxton equation

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Oleg I. Morozov
Keyword(s):  
Lie Algebra ◽  
Spectral Parameter ◽  
Infinite Number ◽  
Higher Order ◽  
Lax Representation ◽  
Recursion Operators ◽  
Higher Symmetries ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Algebra

AbstractWe consider integrability structures of the generalized Hunter–Saxton equation. We obtain the Lax representation with non-removable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra of higher symmetries, and prove existence of infinite number of cosymmetries of higher order. Further, we give examples of employing the higher order symmetries to constructing exact globally defined solutions for the generalized Hunter–Saxton equation.

scholarly journals Symmetry group analysis and invariant solutions of hydrodynamic-type systems

2004 ◽  
Vol 2004 (10) ◽  
pp. 487-534 ◽  
Author(s):  
M. B. Sheftel
Keyword(s):  
Group Analysis ◽  
Type Systems ◽  
Hydrodynamic Type ◽  
Recursion Operators ◽  
Higher Symmetries ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Component Systems ◽  
Two Component Systems ◽  
Existence Conditions

We study point and higher symmetries of systems of the hydrodynamic type with and without an explicit dependence ont,x. We consider such systems which satisfy the existence conditions for an infinite-dimensional group of hydrodynamic symmetries which implies linearizing transformations for these systems. Under additional restrictions on the systems, we obtain recursion operators for symmetries and use them to construct infinite discrete sets of exact solutions of the studied equations. We find the interrelation between higher symmetries and recursion operators. Two-component systems are studied in more detail thann-component systems. As a special case, we consider Hamiltonian and semi-Hamiltonian systems of Tsarëv.

Infinite-Dimensional Lie Algebra

2004 ◽  
pp. 202-202
Author(s):  
John Howie ◽  
Steven Duplij ◽  
Ali Mostafazadeh ◽  
Masaki Yasue ◽  
Vladimir Ivashchuk ◽  
...  
Keyword(s):  
Lie Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Algebra

scholarly journals A COHOMOLOGICAL INTERPRETATION OF THE MIGDAL–MAKEENKO EQUATIONS

2002 ◽  
Vol 17 (08) ◽  
pp. 481-489 ◽  
Author(s):  
A. AGARWAL ◽  
S. G. RAJEEV
Keyword(s):  
Lie Algebra ◽  
Equations Of Motion ◽  
Generating Functional ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Change Of Variables ◽  
Anomalous Term ◽  
Large N ◽  
Infinite Dimensional Lie Algebra ◽  
Cohomological Interpretation

The equations of motion of quantum Yang–Mills theory (in the planar "large-N" limit), when formulated in loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones, i.e. the connection one-form A, to the holonomy U. An infinite-dimensional Lie algebra related to this change of variables (the Lie algebra of loop substitutions) is developed, and the anomaly is interpreted as an element of the first cohomology of this Lie algebra. The Migdal–Makeenko equations are shown to be the condition for the invariance of the Yang–Mills generating functional Z under the action of the generators of this Lie algebra. Connections of this formalism to the collective field approach of Jevicki and Sakita are also discussed.

scholarly journals Matrix 3-Lie superalgebras and BRST supersymmetry

2017 ◽  
Vol 14 (11) ◽  
pp. 1750160 ◽  
Author(s):  
Viktor Abramov
Keyword(s):  
Clifford Algebra ◽  
Lie Algebra ◽  
Vector Fields ◽  
Jacobi Identity ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Algebra Approach ◽  
The Matrix ◽  
Infinite Dimensional Lie Algebra

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

scholarly journals SYMMETRIES AND LIE ALGEBRA OF THE DIFFERENTIAL–DIFFERENCE KADOMSTEV–PETVIASHVILI HIERARCHY

2010 ◽  
Vol 24 (10) ◽  
pp. 1033-1042 ◽  
Author(s):  
XIAN-LONG SUN ◽  
DA-JUN ZHANG ◽  
XIAO-YING ZHU ◽  
DENG-YUAN CHEN
Keyword(s):  
Lie Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Algebra

By introducing suitable non-isospectral flows, we construct two sets of symmetries for the isospectral differential–difference Kadomstev–Petviashvili hierarchy. The symmetries form an infinite dimensional Lie algebra.

Sign in / Sign up

Export Citation Format

Share Document