scholarly journals Trigonometric Real Form of the Spin RS Model of Krichever and Zabrodin

2020 ◽  
Author(s):  
M. Fairon ◽  
L. Fehér ◽  
I. Marshall
Keyword(s):  
Phase Space ◽  
Lie Group ◽  
Poisson Structure ◽  
Hamiltonian Structure ◽  
Equations Of Motion ◽  
Hamiltonian Reduction ◽  
Real Form ◽  
Free System ◽  
Heisenberg Double ◽  
Real Forms

AbstractWe investigate the trigonometric real form of the spin Ruijsenaars–Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable ‘free’ system carried by a spin extension of the Heisenberg double of the $${\mathrm{U}}(n)$$ U ( n ) Poisson–Lie group. The Poisson structure on the unreduced real phase space $${\mathrm{GL}}(n,{\mathbb {C}})\times {\mathbb {C}}^{nd}$$ GL ( n , C ) × C nd is the direct product of that of the Heisenberg double and $$d\ge 2$$ d ≥ 2 copies of a $${\mathrm{U}}(n)$$ U ( n ) covariant Poisson structure on $${\mathbb {C}}^n \simeq {\mathbb {R}}^{2n}$$ C n ≃ R 2 n found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars–Schneider system and we prove its degenerate integrability.

scholarly journals Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

2021 ◽  
Author(s):  
L. Fehér
Keyword(s):  
Lie Group ◽  
Poisson Structure ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Hamiltonian Structures ◽  
Poisson Reduction ◽  
Sutherland Model ◽  
Heisenberg Double ◽  
Real Forms

AbstractWe construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) , which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

scholarly journals A Drift-Free Left Invariant Control System on the Lie Group SO(3)×R3×R3

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Camelia Pop
Keyword(s):  
Control System ◽  
Numerical Integration ◽  
Lie Algebra ◽  
Lie Group ◽  
Poisson Structure ◽  
Dual Space ◽  
Free System ◽  
Geometrical Properties ◽  
Free Left ◽  
Invariant Control

A controllable drift-free system on the Lie group G=SO(3)×R3×R3 is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on g∗,·,·- are studied, where ·,·- is the minus Lie-Poisson structure on the dual space g∗ of the Lie algebra g=so(3)×R3×R3 of G. The numerical integration of this system is also discussed.

scholarly journals QUANTUM ALGEBRA OF THE PARTICLE MOVING ON THE q-DEFORMED MASS-HYPERBOLOID

2000 ◽  
Vol 15 (21) ◽  
pp. 3277-3286
Author(s):  
BOGDAN MORARIU
Keyword(s):  
Phase Space ◽  
Quantum Algebra ◽  
Quantum Phase ◽  
Real Form ◽  
Zero Modes ◽  
Heisenberg Double

I introduce a reality structure on the Heisenberg double of Fun q( SL (N,C)) for q phase, which for N=2 can be interpreted as the quantum phase space of the particle on the q-deformed mass-hyperboloid. This construction is closely related to the q-deformation of the symmetric top. Finally, I conjecture that the above real form describes zero modes of certain noncompact WZNZ-models.

scholarly journals Heisenberg Doubles for Snyder-Type Models

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1055
Author(s):  
Stjepan Meljanac ◽  
Anna Pachoł
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Dual Pair ◽  
One Dimension ◽  
Lorentz Algebra ◽  
Explicit Formulae ◽  
Heisenberg Double

A Snyder model generated by the noncommutative coordinates and Lorentz generators closes a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. This leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.

scholarly journals Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1309
Author(s):  
Jerzy Lukierski
Keyword(s):  
Phase Space ◽  
Hopf Algebras ◽  
Deformation Parameter ◽  
Heisenberg Algebra ◽  
Planck Length ◽  
Covariant Quantum ◽  
Drinfeld Twist ◽  
Quantum Deformations ◽  
Heisenberg Double ◽  
Conformal Symmetries

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed D=4 quantum inhomogeneous conformal Hopf algebras Uθ(su(2,2)⋉T4) and Uθ¯(su(2,2)⋉T¯4), where T4 describes complex twistor coordinates and T¯4 the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of D=4 Heisenberg-conformal algebra (HCA) su(2,2)⋉Hℏ4,4 (Hℏ4,4=T¯4⋉ℏT4 is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length λp, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in Hℏ4,4. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra Uθ(su(2,2)⋉T4).

scholarly journals The Real Forms of the Fractional Supergroup SL(2,C)

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 933
Author(s):  
Yasemen Ucan ◽  
Resat Kosker
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
The Real ◽  
And Mathematics ◽  
Real Forms

The real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group SL2,C is one of these important groups. There are real forms of the classical Lie group SL2,C and the quantum group SL2,C in the literature. Inspired by this, in our study, we obtain the real forms of the fractional supergroups shown with A3NSL2,C, for the non-trivial N = 1 and N = 2 cases, that is, the real forms of the fractional supergroups A31SL2,C and A32SL2,C.

scholarly journals On the Dirac eigenvalues as observables of the on-shell N = 2D = 4 Euclidean supergravity

Open Physics ◽  
2008 ◽  
Vol 6 (4) ◽  
Author(s):  
Ion Vancea
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Gauge Transformations

AbstractWe generalize previous works on the Dirac eigenvalues as dynamical variables of Euclidean gravity and N =1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their value for the Dirac eigenvalues.

scholarly journals Generalized Hamiltonian biodynamics and topology invariants of humanoid robots

2002 ◽  
Vol 31 (9) ◽  
pp. 555-565 ◽  
Author(s):  
Vladimir Ivancevic
Keyword(s):  
Phase Space ◽  
Lie Group ◽  
Humanoid Robots ◽  
And Topology ◽  
Hamiltonian Model

Humanoid robots are anthropomorphic mechanisms with biodynamics that resembles human musculo-skeletal dynamics. This paper proposes a new generalized (dissipative, muscle-driven, stochastic) Hamiltonian model of humanoid biodynamics. Also, (co)homological analysis is performed on its Lie-group based configuration and momentum phase-space manifolds.

PATH INTEGRAL QUANTIZATION OF SUPERSTRING

2010 ◽  
Vol 25 (02) ◽  
pp. 135-141
Author(s):  
H. A. ELEGLA ◽  
N. I. FARAHAT
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Path Integral ◽  
Equations Of Motion ◽  
Singular System ◽  
Constrained Systems ◽  
Classical Structure ◽  
Path Integral Quantization ◽  
Total Differential

Motivated by the Hamilton–Jacobi approach of constrained systems, we analyze the classical structure of a four-dimensional superstring. The equations of motion for a singular system are obtained as total differential equations in many variables. The path integral quantization based on Hamilton–Jacobi approach is applied to quantize the system, and the integration is taken over the canonical phase space coordinates.

Sign in / Sign up

Export Citation Format

Share Document