scholarly journals Mutation-periodic quivers, integrable maps and associated Poisson algebras

2011 ◽  
Vol 369 (1939) ◽  
pp. 1264-1279 ◽  
Author(s):  
Allan P. Fordy
Keyword(s):  
Poisson Bracket ◽  
Canonical Transformation ◽  
Hamiltonian Structure ◽  
Poisson Algebra ◽  
Complete Integrability ◽  
Canonical Coordinates ◽  
Poisson Algebras ◽  
Invariant Functions ◽  
Liouville’S Equation

We consider a class of map, recently derived in the context of cluster mutation. In this paper, we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson-commuting functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility of a pair of these functions gives rise to Liouville’s equation and the map plays the role of a Bäcklund transformation.

scholarly journals REDUCTION OF TODA LATTICE HIERARCHY TO GENERALIZED KdV HIERARCHIES AND THE TWO-MATRIX MODEL

1995 ◽  
Vol 10 (17) ◽  
pp. 2537-2577 ◽  
Author(s):  
H. ARATYN ◽  
E. NISSIMOV ◽  
S. PACHEVA ◽  
A.H. ZIMERMAN
Keyword(s):  
Poisson Bracket ◽  
Hamiltonian Structure ◽  
Toda Lattice ◽  
Free Field ◽  
String Model ◽  
Matrix Formulation ◽  
Hamiltonian Action ◽  
Kdv Hierarchy ◽  
Associated Matrix ◽  
Toda Lattice Hierarchy

Toda lattice hierarchy and the associated matrix formulation of the 2M-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which Abelianize the second KP Hamiltonian structure, we are able to obtain a unified formalism for the reduced SL (M+1, M−k) KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded SL (M+1, M−k) matrices in the diagonal gauge and we describe their bracket structure and field content. In particular, we provide explicit free field representations of the associated W(M, M−k) Poisson bracket algebras generalizing the familiar nonlinear WM+1 algebra. Discrete Bäcklund transformations for SL (M+1, M−k) KdV are generated naturally from lattice translations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the SL (M+1, 1) KdV hierarchy.

scholarly journals Nambu–Poisson bracket on superspace

2018 ◽  
Vol 15 (11) ◽  
pp. 1850190 ◽  
Author(s):  
Viktor Abramov
Keyword(s):  
Poisson Bracket ◽  
Jacobi Identity ◽  
Leibniz Rule ◽  
Smooth Functions ◽  
Poisson Algebras ◽  
Fundamental Identity ◽  
Odd Degree ◽  
New Formula ◽  
Usual Formula

We propose an extension of [Formula: see text]-ary Nambu–Poisson bracket to superspace [Formula: see text] and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace [Formula: see text]. We prove in the case of the superspaces [Formula: see text] and [Formula: see text] that our [Formula: see text]-ary bracket, defined with the help of superdeterminant, satisfies the conditions for [Formula: see text]-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of [Formula: see text]-ary bracket defined with the help of superdeterminant in the case of superspace [Formula: see text] and show that it is the sum of usual [Formula: see text]-ary Nambu–Poisson bracket and a new [Formula: see text]-ary bracket, which we call [Formula: see text]-bracket, where [Formula: see text] is the product of two odd degree smooth functions.

scholarly journals Quantization of a Poisson algebra and polynomials associated to links

1990 ◽  
Vol 120 ◽  
pp. 113-127 ◽  
Author(s):  
Tetsuya Ozawa
Keyword(s):  
Riemann Surface ◽  
Lie Algebra ◽  
Poisson Algebra ◽  
Homotopy Classes ◽  
Poisson Algebras

A formal quantization of Poisson algebras was discussed by several authors (see for instance Drinfel’d [D]). A formal Lie algebra generated by homotopy classes of loops on a Riemann surface ∑ was obtained by W. Goldman in [G], and its Poisson algebra was quantized, in the sense of Drinfel’d, by Turaev in [T].

Anatomy of the canonical transformation

1993 ◽  
Vol 345 (1677) ◽  
pp. 577-598 ◽  
Keyword(s):  
Fluid Dynamics ◽  
Canonical Transformation ◽  
Generating Functions ◽  
Hamiltonian Structure ◽  
Rigid Bodies ◽  
Symplectic Integrators ◽  
Classical Dynamics ◽  
Canonical Transformations ◽  
Geophysical Fluid Dynamics ◽  
Dynamics Of Particles

A concise account of the structure of the canonical transformation is given, in the lowest dimensional case. This case is chosen because it offers a special clarity in several respects. In particular, the diversity of possible generating functions is illustrated by m any examples which are not available elsewhere. Many of these are of physical interest, and some of them are multivalued. These examples are used to inform a comparative study of the several different definitions of a canonical transformation to be found in the literature. The paper is pertinent to all those branches of mechanics which can be given a hamiltonian representation. These include not only the classical dynamics of particles and rigid bodies, but also some more recent studies in continuum mechanics, including geophysical fluid dynamics. An area of particular modern interest is that of symplectic integrators. These are numerical integrating algorithms which generate a solution to Hamilton’s equations via a sequence of canonical transformations, which preserve the hamiltonian structure in the numerical solution.

scholarly journals MINISUPERSPACES: OBSERVABLES AND QUANTIZATION

1993 ◽  
Vol 02 (01) ◽  
pp. 15-50 ◽  
Author(s):  
ABHAY ASHTEKAR ◽  
RANJEET TATE ◽  
CLAES UGGLA
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
Canonical Transformation ◽  
Parameter Family ◽  
Hamiltonian Constraint ◽  
Canonical Coordinates ◽  
Quantum Theories ◽  
Canonical Variables ◽  
Unitary Transformations ◽  
Homogeneous Cosmologies

A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e. phase-space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through “deparametrization” and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory in spite of the fact that the evolution is implemented by a one-parameter family of unitary transformations. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to the quantum theory following an independent avenue. The two quantum theories — based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables — are compared and shown to be equivalent.

scholarly journals Universal enveloping Hom-algebras of regular Hom-Poisson algebras

2022 ◽  
Vol 7 (4) ◽  
pp. 5712-5727
Author(s):  
Xianguo Hu ◽  
Keyword(s):  
Poisson Algebra ◽  
Poisson Algebras

<abstract><p>In this paper, we introduce universal enveloping Hom-algebras of Hom-Poisson algebras. Some properties of universal enveloping Hom-algebras of regular Hom-Poisson algebras are discussed. Furthermore, in the involutive case, it is proved that the category of involutive Hom-Poisson modules over an involutive Hom-Poisson algebra $ A $ is equivalent to the category of involutive Hom-associative modules over its universal enveloping Hom-algebra $ U_{eh}(A) $.</p></abstract>

scholarly journals SECOND-ORDER DEFORMATIONS OF HYDRODYNAMIC-TYPE POISSON BRACKETS

2009 ◽  
Vol 51 (A) ◽  
pp. 75-82
Author(s):  
JAMES T. FERGUSON
Keyword(s):  
Differential Operator ◽  
Poisson Bracket ◽  
Hamiltonian Structure ◽  
Second Order ◽  
Hydrodynamic Type ◽  
Poisson Brackets ◽  
Geometric Type

AbstractThis paper is concerned with the properties of differential-geometric-type Poisson brackets specified by a differential operator of degree 2. It also considers the conditions required for such a Poisson bracket to form a bi-Hamiltonian structure with a hydrodynamic-type Poisson bracket.

scholarly journals SYMMETRIES AND INVARIANTS OF TWISTED QUANTUM ALGEBRAS AND ASSOCIATED POISSON ALGEBRAS

2008 ◽  
Vol 20 (02) ◽  
pp. 173-198 ◽  
Author(s):  
A. I. MOLEV ◽  
E. RAGOUCY
Keyword(s):  
Poisson Bracket ◽  
Quantum Algebras ◽  
Poisson Algebras ◽  
Triangular Matrices ◽  
Polynomial Invariants ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebras ◽  
Quantized Enveloping Algebra ◽  
Group B ◽  
Mathematical Literature

We construct an action of the braid group BN on the twisted quantized enveloping algebra [Formula: see text] where the elements of BN act as automorphisms. In the classical limit q → 1, we recover the action of BN on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra [Formula: see text]. We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras.

scholarly journals Products of twists, geodesic lengths and Thurston shears

2014 ◽  
Vol 151 (2) ◽  
pp. 313-350 ◽  
Author(s):  
Scott A. Wolpert
Keyword(s):  
Poisson Bracket ◽  
Symplectic Geometry ◽  
Poisson Algebra ◽  
Weighted Sum ◽  
Hyperbolic Surfaces ◽  
Shear Deformations ◽  
Geodesic Laminations ◽  
Coordinate Algebra ◽  
Left And Right

AbstractThurston introduced shear deformations (cataclysms) on geodesic laminations–deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined and the Weil–Petersson (WP) duality of shears and the defined length is established. The Poisson bracket of a pair of balanced weight systems on a set of disjoint ideal geodesics is given in terms of an elementary$2$-form. The symplectic geometry of balanced weight systems on ideal geodesics is developed. Equality of the Fock shear coordinate algebra and the WP Poisson algebra is established. The formula for the WP Riemannian pairing of shears is also presented.

scholarly journals On (co)homology of Frobenius Poisson algebras

2014 ◽  
Vol 14 (2) ◽  
pp. 371-386 ◽  
Author(s):  
Can Zhu ◽  
Fred Van Oystaeyen ◽  
Yinhuo Zhang
Keyword(s):  
Bilinear Form ◽  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Poisson Algebra ◽  
Hochschild Homology ◽  
Poisson Algebras ◽  
Poisson Homology ◽  
Poisson Cohomology ◽  
Frobenius Algebras

AbstractIn this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the non-degenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a Batalin-Vilkovisky structure on the Poisson cohomology ring making it into a Batalin-Vilkovisky algebra.

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