scholarly journals COMPATIBLE POISSON STRUCTURES OF TODA TYPE DISCRETE HIERARCHY

2005 ◽  
Vol 20 (07) ◽  
pp. 1367-1388 ◽  
Author(s):  
HENRIK ARATYN ◽  
KLAUS BERING
Keyword(s):  
Poisson Structure ◽  
Integrable Hierarchy ◽  
Difference Operators ◽  
Baxter Equation ◽  
Poisson Structures ◽  
Algebra Isomorphism ◽  
R Matrix ◽  
Compatible Poisson Structures ◽  
Special Value

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and nonlocal families of R-matrix solutions to the modified Yang–Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.

scholarly journals Degenerate Sklyanin algebras

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Andrey Smirnov
Keyword(s):  
Difference Operators ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Rational Solutions ◽  
R Matrix ◽  
Gauge Transformations ◽  
Sklyanin Algebras ◽  
Sklyanin Algebra ◽  
The Difference

AbstractNew trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.

GAUGE FIXING AND QUANTUM GROUP SYMMETRIES IN (2+1)-GRAVITY

2013 ◽  
Vol 10 (08) ◽  
pp. 1360004
Author(s):  
CATHERINE MEUSBURGER ◽  
TORSTEN SCHÖNFELD
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
Baxter Equation ◽  
Poisson Structures ◽  
R Matrix ◽  
Point Particles ◽  
Combinatorial Description ◽  
Gauge Fixing ◽  
Group Symmetries ◽  
Chern Simons

We summarize the results obtained by applying Dirac's gauge fixing formalism to the combinatorial description of the Chern–Simons formulation of (2+1)-gravity and their implications for the symmetries of the quantum theory. While the combinatorial description of the phase space exhibits standard Poisson–Lie symmetries, every gauge fixing condition based on two point particles yields a Poisson structure determined by a dynamical classical r-matrix. By considering transformations between different gauge fixing conditions, it is possible to classify all gauge fixed Poisson structures in terms of two standard solutions of the dynamical classical Yang–Baxter equation. We discuss the conclusions that can be drawn from this about the symmetries of (2+1)-dimensional quantum gravity.

scholarly journals Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

2021 ◽  
Author(s):  
Alexey V. Bolsinov ◽  
Andrey Yu. Konyaev ◽  
Vladimir S. Matveev
Keyword(s):  
Singular Points ◽  
Recursion Operator ◽  
Poisson Structures ◽  
Degeneracy Condition ◽  
Compatible Poisson Structures

AbstractWe study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

Geometry of Poisson Structures

1995 ◽  
Vol 2 (4) ◽  
pp. 347-359
Author(s):  
Z. Giunashvili
Keyword(s):  
Poisson Structure ◽  
Poisson Structures ◽  
Geometrical Structures

Abstract The purpose of this paper is to consider certain mechanisms of the emergence of Poisson structures on a manifold. We shall also establish some properties of the bivector field that defines a Poisson structure and investigate geometrical structures on the manifold induced by such fields. Further, we shall touch upon the dualism between bivector fields and differential 2-forms.

Transfer Matrix of the Two-dimensional Ising Model

2020 ◽  
pp. 211-234
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Ising Model ◽  
Critical Point ◽  
Transfer Matrix ◽  
Functional Equations ◽  
Baxter Equation ◽  
Matrix Formalism ◽  
Two Dimensional ◽  
R Matrix ◽  
Transfer Matrix Formalism ◽  
Lowest Eigenvalue

This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point can be identified by means of the lowest eigenvalue. The chapter covers Baxter's approach, the Yang–Baxter equation and its relation to the Boltzmann weights, the R-matrix, and discusses activity away from the critical point, the six-vertex model, as well as functional equations and symmetries.

Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

2014 ◽  
Vol 6 (01) ◽  
pp. 87-106
Author(s):  
Xueyang Li ◽  
Aiguo Xiao ◽  
Dongling Wang
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Generating Function ◽  
Poisson Structure ◽  
Poisson Structures ◽  
Volterra Systems

AbstractThe generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices. In this paper, we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices. In particular, some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems (such as generalized Lotka-Volterra systems, Robbins equations and so on).

FERMIONS AND LINK INVARIANTS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 493-532 ◽  
Author(s):  
L. Kauffman ◽  
H. Saleur
Keyword(s):  
Braid Group ◽  
Degrees Of Freedom ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Group Representations ◽  
R Matrix ◽  
Link Invariants ◽  
Linear Representations ◽  
State Models

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

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.

Spectrum-generating bialgebra of the hexagonal-lattice dimer model

2009 ◽  
Vol 87 (10) ◽  
pp. 1099-1125 ◽  
Author(s):  
Jeffrey R. Schmidt
Keyword(s):  
Hexagonal Lattice ◽  
Baxter Equation ◽  
Dimer Model ◽  
R Matrix ◽  
The Matrix ◽  
Complete Set

The method of constructing a complete set of “zero-operator” identities preserved by the matrix coproduct is shown to be general, and is used to build the operator bialgebra for the hexagonal-lattice dimer model. This technique is complementary to the RTT-equation, and does not require a solution to the Yang–Baxter equation. The resulting bialgebra in the case of hexagonal lattice dimers has a distinctly Yangian structure, but has no R-matrix or antipode.

Sign in / Sign up

Export Citation Format

Share Document