ALGEBRAIC STUDY ON THE POISSON STRUCTURE OF LAX OPERATOR OF B AND C TYPE AND SUPER LAX OPERATOR.

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 405-418
Author(s):  
KAORU IKEDA
Keyword(s):  
Poisson Structure ◽  
Algebraic Structure ◽  
Lax Equation ◽  
Lax Operator

The Poisson structure of Lax operator of B and C type and super Lax operator which has odd parity are studied. The algebraic structure of Poisson structure as a background of Lax equation is thrown light on.

CANONICAL BÄCKLUND TRANSFORMATION FOR A DISCRETE INTEGRABLE CHAIN AND ITS ASSOCIATED PROPERTIES

2003 ◽  
Vol 18 (16) ◽  
pp. 1127-1139
Author(s):  
A. GHOSE CHOUDHURY ◽  
BARUN KHANRA ◽  
A. ROY CHOWDHURY
Keyword(s):  
Poisson Structure ◽  
Algebraic Structure ◽  
Bäcklund Transformation ◽  
Inverse Transformation ◽  
Strict Sense ◽  
Backlund Transformation ◽  
Lax Equation ◽  
R Matrix ◽  
Lax Operator ◽  
Time Part

The concept of a canonical Bäcklund transformation as laid down by Sklyanin is extended to a discrete integral chain, with a Poisson structure which is not canonical in the strict sense. The transformation is induced by an auxiliary Lax operator with a classical r-matrix which is similar in its algebraic structure to that of the original Lax operator governing the dynamics of the chain. Moreover, the transformation can be obtained from a suitable generating function. It is also shown how successive transformations can be composed to construct a new transformation. Finally an inverse transformation is also constructed. The compatibility of the transformation with the "time" part of the Lax equation is explicitly demonstrated. It is also shown that the Bianchi theorem of permutability holds good.

POISSON STRUCTURE AND INTEGRABILITY OF THE NEUMANN-MOSER-UHLENBECK MODEL

1990 ◽  
Vol 05 (23) ◽  
pp. 4477-4488 ◽  
Author(s):  
J. AVAN ◽  
M. TALON
Keyword(s):  
Poisson Structure ◽  
Point Of View ◽  
Conserved Quantities ◽  
Poisson Brackets ◽  
Quantum Level ◽  
Quantum Integrability ◽  
Commuting Operators ◽  
Lax Operator

Neumann’s model, describing the motion of a particle on an N-sphere under harmonic forces, is studied from the point of view of classical and quantum integrability. Classical integrability is derived from a generalized structure, “R-S couple” or “D-matrix” for the Poisson brackets of the Lax operator. The already-known set of conserved quantities for this model turns out to follow straightforwardly from this structure. It gives rise to a set of commuting operators at the quantum level, and the algebra of Lax operators directly follows from the classical one.

scholarly journals LIE-POISSON STRUCTURE OF CONFORMAL NON-ABELIAN THIRRING MODELS

1994 ◽  
Vol 09 (06) ◽  
pp. 483-489 ◽  
Author(s):  
OLEG A. SOLOVIEV
Keyword(s):  
Hamiltonian Systems ◽  
Poisson Structure ◽  
Algebraic Structure ◽  
Coupling Constants ◽  
Poisson Brackets ◽  
Quantum Level ◽  
Hamiltonian Method ◽  
Special Values ◽  
Lie Algebraic Structure ◽  
Hamiltonian Quantization

It is shown that non-Abelian Thirring models can be formulated as the Hamiltonian systems with Poisson brackets of the Lie algebraic structure. This fact allows Thirring models to be quantized by the Hamiltonian method. We show that the classical Lie-Poisson structure can be promoted to the quantum level in two different ways corresponding to different phases of non-Abelian Thirring models. There are special values of coupling constants at which the Hamiltonian quantization of Thirring models can be carried out consistently with the conformal invariance. These fixed couplings appear to be the solutions of the Virasoro master equation.

scholarly journals SUPER-LAX OPERATOR EMBEDDED IN SELF-DUAL SUPERSYMMETRIC YANG-MILLS THEORY

1996 ◽  
Vol 11 (30) ◽  
pp. 2417-2426 ◽  
Author(s):  
HITOSHI NISHINO
Keyword(s):  
Integrable Models ◽  
Three Dimensions ◽  
Superstring Theory ◽  
Lax Equation ◽  
Yang Mills ◽  
Guiding Principle ◽  
Lax Operator ◽  
Geometrical Relationship

We show that the super-Lax operator for N=1 supersymmetric Kadomtsev-Petviashvili (SKP) equation of Manin and Radul in three dimensions can be embedded into recently developed self-dual supersymmetric Yang-Mills theory in 2+2 dimensions, based on general features of its underlying super-Lax equation. The whole hierarchy of the SKP equations of Manin and Radul is generated by geometrical superfield equations of self-dual supersymmetric Yang-Mills theory. The differential geometrical relationship in superspace between the embedding principle of the super-Lax operator and its associated super-Sato equation is clarified. This result provides a good guiding principle for the embedding of other integrable subsystems in the super-Lax equation into the four-dimensional self-dual supersymmetric Yang-Mills theory, which is the consistent background for N=2 superstring theory, and potentially generates other unknown supersymmetric integrable models in lower dimensions.

scholarly journals Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

2019 ◽  
Vol 6 (1) ◽  
pp. 88-102
Author(s):  
Yat Sun Poon ◽  
John Simanyi
Keyword(s):  
Poisson Structure ◽  
Algebraic Structure ◽  
Complex Structure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Poisson Cohomology ◽  
Abelian Complex Structure ◽  
Gerstenhaber Algebras ◽  
Holomorphic Poisson Cohomology ◽  
Necessary And Sufficient

AbstractIt is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure.We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.

scholarly journals Gauged double field theory as an L∞ algebra

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Eric Lescano ◽  
Martín Mayo
Keyword(s):  
Field Theory ◽  
Algebraic Structure ◽  
Classical Field ◽  
Field Theories ◽  
Lie Derivative ◽  
Linear Perturbation ◽  
Double Field Theory ◽  
Generalized Frame ◽  
Symmetry Transformations ◽  
Classical Field Theories

Abstract L∞ algebras describe the underlying algebraic structure of many consistent classical field theories. In this work we analyze the algebraic structure of Gauged Double Field Theory in the generalized flux formalism. The symmetry transformations consist of a generalized deformed Lie derivative and double Lorentz transformations. We obtain all the non-trivial products in a closed form considering a generalized Kerr-Schild ansatz for the generalized frame and we include a linear perturbation for the generalized dilaton. The off-shell structure can be cast in an L3 algebra and when one considers dynamics the former is exactly promoted to an L4 algebra. The present computations show the fully algebraic structure of the fundamental charged heterotic string and the $$ {L}_3^{\mathrm{gauge}} $$ L 3 gauge structure of (Bosonic) Enhanced Double Field Theory.

scholarly journals Polymorphic Iterable Sequential Effect Systems

2021 ◽  
Vol 43 (1) ◽  
pp. 1-79
Author(s):  
Colin S. Gordon
Keyword(s):  
Algebraic Structure ◽  
Position Effect ◽  
Type Systems ◽  
Sequential Effect ◽  
Prior Work ◽  
Sequential Effects ◽  
Wide Range ◽  
The Relationship ◽  
Categorical Semantics

Effect systems are lightweight extensions to type systems that can verify a wide range of important properties with modest developer burden. But our general understanding of effect systems is limited primarily to systems where the order of effects is irrelevant. Understanding such systems in terms of a semilattice of effects grounds understanding of the essential issues and provides guidance when designing new effect systems. By contrast, sequential effect systems—where the order of effects is important—lack an established algebraic structure on effects. We present an abstract polymorphic effect system parameterized by an effect quantale—an algebraic structure with well-defined properties that can model the effects of a range of existing sequential effect systems. We define effect quantales, derive useful properties, and show how they cleanly model a variety of known sequential effect systems. We show that for most effect quantales, there is an induced notion of iterating a sequential effect; that for systems we consider the derived iteration agrees with the manually designed iteration operators in prior work; and that this induced notion of iteration is as precise as possible when defined. We also position effect quantales with respect to work on categorical semantics for sequential effect systems, clarifying the distinctions between these systems and our own in the course of giving a thorough survey of these frameworks. Our derived iteration construct should generalize to these semantic structures, addressing limitations of that work. Finally, we consider the relationship between sequential effects and Kleene Algebras, where the latter may be used as instances of the former.

scholarly journals Two semigroups of continuous relations

1977 ◽  
Vol 23 (1) ◽  
pp. 46-58 ◽  
Author(s):  
A. R. Bednarek ◽  
Eugene M. Norris
Keyword(s):  
Large Class ◽  
Algebraic Structure ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Topological Semigroups ◽  
Stone Type

SynopsisIn this paper we define two semigroups of continuous relations on topological spaces and determine a large class of spaces for which Banach-Stone type theorems hold, i.e. spaces for which isomorphism of the semigroups implies homeomorphism of the spaces. This class includes all 0-dimensional Hausdorff spaces and all those completely regular Hausdorff spaces which contain an arc; indeed all of K. D. Magill's S*-spaces are included. Some of the algebraic structure of the semigroup of all continuous relations is elucidated and a method for producing examples of topological semigroups of relations is discussed.

scholarly journals Equipping weak equivalences with algebraic structure

2019 ◽  
Vol 294 (3-4) ◽  
pp. 995-1019 ◽  
Author(s):  
John Bourke
Keyword(s):  
Algebraic Structure
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