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.

On Automorphisms and Derivations of a Lie Algebra

2006 ◽  
Vol 13 (01) ◽  
pp. 119-132 ◽  
Author(s):  
V. R. Varea ◽  
J. J. Varea
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Nilpotent Lie Algebra ◽  
Identity Component ◽  
Trivial Center

We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.

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 NOISE AND DISSIPATION IN QUANTUM THEORY

1990 ◽  
Vol 02 (02) ◽  
pp. 127-176 ◽  
Author(s):  
LUIGI ACCARDI
Keyword(s):  
Quantum Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Langevin Equation ◽  
Equations Of Motion ◽  
Natural Generalization ◽  
Usual Sense ◽  
Heisenberg Equation ◽  
Generalized Langevin Equation ◽  
Solid State Physics

A model independent generalization of quantum mechanics, including the usual as well as the dissipative quantum systems, is proposed. The theory is developed deductively from the basic principles of the standard quantum theory, the only new qualitative assumption being that we allow the wave operator at time t of a quantum system to be non-differentiable (in t) in the usual sense, but only in an appropriately defined (Sec. 5) stochastic sense. The resulting theory is shown to lead to a natural generalization of the usual quantum equations of motion, both in the form of the Schrödinger equation in interaction representation (Sec. 6) and of the Heisenberg equation (Sec. 8). The former equation leads in particular to a quantum fluctuation-dissipation relation of Einstein’s type. The latter equation is a generalized Langevin equation, from which the known form of the generalized master equation can be deduced via the quantum Feynmann-Kac technique (Secs. 9 and 10). For quantum noises with increments commuting with the past the quantum Langevin equation defines a closed system of (usually nonlinear) stochastic differential equations for the observables defining the coefficients of the noises. Such systems are parametrized by certain Lie algebras of observables of the system (Sec. 10). With appropriate choices of these Lie algebras one can deduce generalizations and corrections of several phenomenological equations previously introduced at different times to explain different phenomena. Two examples are considered: the Lie algebra [q, p]=i (Sec. 12), which is shown to lead to the equations of the damped harmonic oscillator; and the Lie algebra of SO(3) (Sec. 13) which is shown to lead to the Bloch equations. In both cases the equations obtained are independent of the model of noise. Moreover, in the former case, it is proved that the only possible noises which preserve the commutation relations of p, q are the quantum Brownian motions, commonly used in laser theory and solid state physics.

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.

scholarly journals A pure connection formulation with real fields for gravity

2020 ◽  
pp. 2050114
Author(s):  
J. E. Rosales-Quintero
Keyword(s):  
Lie Algebra ◽  
Lagrange Multipliers ◽  
General Class ◽  
Equations Of Motion ◽  
Conformally Flat ◽  
Einstein Manifolds ◽  
De Sitter ◽  
Killing Form ◽  
Dual Approach ◽  
Four Dimensions

We study an [Formula: see text] pure connection formulation in four dimensions for real-valued fields, inspired by the Capovilla, Dell and Jacobson complex self-dual approach. By considering the CMPR BF action, also, taking into account a more general class of the Cartan–Killing form for the Lie algebra [Formula: see text] and by refining the structure of the Lagrange multipliers, we integrate out the metric variables in order to obtain the pure connection action. Once we have obtained this action, we impose certain restrictions on the Lagrange multipliers, in such a way that the equations of motion led us to a family of torsionless conformally flat Einstein manifolds, parametrized by two numbers. Finally, we show that, by a suitable choice of parameters, self-dual spaces (Anti-) de Sitter can be obtained.

scholarly journals Classical Integrability of Chiral QCD2 and Classical Curves

1997 ◽  
Vol 12 (32) ◽  
pp. 2445-2453 ◽  
Author(s):  
Robert De Mello Koch ◽  
João P. Rodrigues
Keyword(s):  
Large Class ◽  
Gauge Group ◽  
Infinite Number ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Conserved Quantities ◽  
Large N ◽  
Fermionic Fields

In this letter, classical chiral QCD 2 is studied in the lightcone gauge A-=0. The once integrated equation of motion for the current is shown to be of the Lax form, which demonstrates an infinite number of conserved quantities. Specializing to gauge group SU(2), we show that solutions to the classical equations of motion can be identified with a very large class of curves. We demonstrate this correspondence explicitly for two solutions. The classical fermionic fields associated with these currents are then obtained. Finally, we conclude by showing how 't Hooft's large-N solution is obtained from one of our solutions.

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.

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