Cayley--Klein Poisson Homogeneous Spaces

Herranz, Francisco J.; Ballesteros, Ángel; Gutiérrez-Sagredo, Iván; Santander, Mariano

doi:10.7546/giq-20-2019-161-183

Cited by 5 publications

(19 citation statements)

References 39 publications

(109 reference statements)

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“…In the procedure that makes use of the real parameters (ω 1 , ω 2 ) [7][8][9][10][11][12], the generic CK Lie algebra is denoted by so ω 1 ,ω 2 (3) = span{J 01 , J 02 , J 12 }, which corresponds to a two-parametric family of Lie algebras with commutation relations given by [J 12 (1)…”

Section: Ellipticmentioning

confidence: 99%

“…In addition, we stress that the same CK group SO(2, 1) appears three times in Table 1, and two of their CK spaces are "similar" (see [15,16] for a very detailed description of these three geometries in terms of hypercomplex numbers). The structure of the three CK geometries involved, hyperbolic and the (anti-)de Sitter ones, can be better understood by considering not only the usual CK space (3) shown in Table 1 but also their 2D homogeneous spaces of lines, as it was performed in [8,12] and likewise for the Euclidean and Poincaré groups, which appear twice in Table 1.…”

Section: Ellipticmentioning

confidence: 99%

“…which is a solution of the modified classical Yang-Baxter Equation ( 14) (its Schouten bracket does not vanish now), and thus r D defines a quasitriangular or standard quantum deformation of g with a coboundary Lie bialgebra (g, δ D (r D )) determined by a cocommutator δ D through the relation (12). Concerning the Drinfel'd-Jimbo quantum deformations of semisimple algebras [28][29][30], it is known that they are closely related to quantum deformations of Drinfel'd doubles, that is, quantum doubles, in such a manner that they are "almost" but not strictly speaking quantum doubles [24].…”

Section: Drinfel'd Double Structuresmentioning

confidence: 99%

“…In order to set up the main ideas and the formalism that we shall follow along the whole paper, let us consider the well-known nine 2D CK geometries [3,[5][6][7][8][9][10][11][12]. These emerge as the different possibilities for considering the measures of distance between two points and the measure of an angle between two lines, with each of them being either of elliptic, parabolic or hyperbolic type.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, there are three possible kind of hypercomplex numbers according to the specific unit ι: (1) If ι 2 = −1, then ι is an elliptical unit providing the usual complex numbers; (2) If ι 2 = +1, ι is a hyperbolic unit yielding the so-called split complex, double or Clifford numbers; and (3) if ι 2 = 0, ι is a parabolic unit leading to the dual or Study numbers. Alternatively, 2D CK geometries can also be studied in terms of two real graded contraction parameters ω 1 and ω 2 , which can take positive, negative or zero values [7][8][9][10][11][12]. By taking into account the above two approaches, we display the specific 2D CK geometries in Table 1, where they are named in their original geometric form [3], as well as in their physical (or kinematical) terminology (second and third rows).…”

Section: Introductionmentioning

confidence: 99%

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Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

Gutiérrez-Sagredo

Herranz

2021

Symmetry

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The Cayley–Klein (CK) formalism is applied to the real algebra so(5) by making use of four graded contraction parameters describing, in a unified setting, 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel’d–Jimbo real Lie bialgebra for so(5) together with its Drinfel’d double structure, we obtain the corresponding CK bialgebra and the CK r-matrix coming from a Drinfel’d double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring dealing with real structures, we found that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we found 14 classical r-matrices coming from Drinfel’d doubles, obtaining new results for the de Sitter so(4,1) and anti-de Sitter so(3,2) as well as for some of their contractions. These geometric results were exhaustively applied onto the (3 + 1)D kinematical algebras, considering not only the usual (3 + 1)D spacetime but also the 6D space of lines. We established different assignations between the geometrical CK generators and the kinematical ones, which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature Λ and the speed of light c. We, finally, obtained four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all κ-deformations as particular cases.

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Section: Ellipticmentioning

confidence: 99%

Section: Ellipticmentioning

confidence: 99%

Section: Drinfel'd Double Structuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

Gutiérrez-Sagredo

Herranz

2021

Symmetry

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Study of information entropy for involved quantum models in complex Cayley-Klein space

2020

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In the language of the complex formalism, we study the information entropy of a particle on the motion groups from a family of the unitary Cayley-Klein space with constant curvature κ. Hence, in making use of the constant curvature, all the results here presented will be simultaneously valid for the 2D coset space SUκ(2)/U(1) in forms of 2D sphere , hyperbolic plane and Euclidean plane . In addition to physical complex coordinates , their corresponding components are expressed in a 2D complex formalism in terms of the constant curvature in Cayley-Klein space. This process enables us to derive information entropies located in the circular well on two spaces, which are the basis for achieving the relationship between Shannon entropy and Fisher information and the inequalities among them. In particular, we notice that the particle has freely behavior in a spherical state , while it behaves as a constraint particle in the situation of the hyperbolic plane .

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All noncommutative spaces of κ-Poincaré geodesics

Ballesteros¹,

Gutiérrez-Sagredo²,

Herranz³

2022

J. Phys. A: Math. Theor.

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Noncommutative spaces of geodesics provide an alternative way of introducing noncommutative relativistic kinematics endowed with quantum group symmetry. In this paper we present explicitly the seven noncommutative spaces of time-, space- and light-like geodesics that can be constructed from the time-, space- and light- versions of the κ-Poincaré quantum symmetry in (3+1) dimensions. Remarkably enough, only for the light-like (or null-plane) κ-Poincaré deformation the three types of noncommutative spaces of geodesics can be constructed, while for the time-like and space-like deformations both the quantum time-like and space-like geodesics can be deﬁned, but not the light-like one. This obstruction comes from the constraint imposed by the coisotropy condition for the corresponding deformation with respect to the isotropy subalgebra associated to the given space of geodesics, since all these quantum spaces are constructed as quantizations of the corresponding classical coisotropic Poisson homogeneous spaces. The known quantum space of geodesics on the light cone is given by a ﬁve-dimensional homogeneous quadratic algebra, and the six nocommutative spaces of time-like and space-like geodesics are explicitly obtained as six-dimensional nonlinear algebras. Five out of these six spaces are here presented for the ﬁrst time, and Darboux generators for all of them are found, thus showing that the quantum deformation parameter κ-1 plays exactly the same algebraic role on quantum geodesics as the Planck constant h plays in the usual phase space description of quantum mechanics.

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Cayley--Klein Poisson Homogeneous Spaces

Cited by 5 publications

References 39 publications

Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

Study of information entropy for involved quantum models in complex Cayley-Klein space

All noncommutative spaces of κ-Poincaré geodesics

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