Drinfel’d double structures for Poincaré and Euclidean groups

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

doi:10.1088/1742-6596/1194/1/012041

Cited by 4 publications

(6 citation statements)

References 29 publications

(80 reference statements)

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“…It can be shown by an explicit calculation that (6.23) satisfies the Fock-Rosly conditions. The r-matrices associated with all possible Drinfeld double structures on the D = 3 Poincaré algebra were recently classified in [18,35], where it was found that there are eight inequivalent D = 3 Poincaré Drinfeld doubles. In terms of the Stachura classification from subsection 5.2, four of them lead to the r-matrices of the type r 6 with = ±1 (and appropriate θ µν ), two have the r-matrix of the type r 2 with = β = 1 (and appropriate θ, θ ), and the remaining two Drinfeld double r-matrices are of the types r 1 with β = 1 and r 7 , respectively.…”

Section: Jhep09(2020)096mentioning

confidence: 99%

Quantum D = 3 Euclidean and Poincaré symmetries from contraction limits

Kowalski-Glikman

Lukierski

Trześniewski

2020

J. High Energ. Phys.

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Following the recently obtained complete classification of quantum-deformed $$ \mathfrak{o} $$ o (4), $$ \mathfrak{o} $$ o (1, 3) and $$ \mathfrak{o} $$ o (2) algebras, characterized by classical r-matrices, we study their inhomogeneous D = 3 quantum IW contractions (i.e. the limit of vanishing cosmological constant), with Euclidean or Lorentzian signature. Subsequently, we compare our results with the complete list of D = 3 inhomogeneous Euclidean and D = 3 Poincaré quantum deformations obtained by P. Stachura. It turns out that the IW contractions allow us to recover all Stachura deformations. We further discuss the applicability of our results in the models of 3D quantum gravity in the Chern-Simons formulation (both with and with- out the cosmological constant), where it is known that the relevant quantum deformations should satisfy the Fock-Rosly conditions. The latter deformations in part of the cases are associated with the Drinfeld double structures, which also have been recently investigated in detail.

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Section: Jhep09(2020)096mentioning

confidence: 99%

Quantum D = 3 Euclidean and Poincaré symmetries from contraction limits

Kowalski-Glikman

Lukierski

Trześniewski

2020

J. High Energ. Phys.

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“…In contrast, as we commented at the end of Section 2.3, in lower dimensions, such structures do exist and the classification of Drinfel'd doubles was recently performed for the (2 + 1)D Poincaré [127] and 3D Euclidean algebras [128]. Moreover, to the best of our knowledge, the classification of Drinfel'd doubles for the (anti-)de Sitter algebras has only been carried out in (2 + 1) dimensions [115].…”

Section: Drinfel'd Double Structures For Cayley-klein Algebrasmentioning

confidence: 94%

“…We recall that the classifications of non-isomorphic 4D and 6D real Drinfel'd double structures were carried out in [124] and [125], respectively, while their Hopf algebra quantizations were constructed in [126]. From these results, and also from [54], there were obtained the classifications of Drinfel'd double structures for the (2 + 1)D (anti-)de Sitter algebras in [115], (2 + 1)D Poincaré algebra and centrally extended (1 + 1)D Poincaré algebra in [127] and 3D Euclidean algebra in [128].…”

Section: Drinfel'd Double Structuresmentioning

confidence: 99%

“…As far as the non-relativistic limit c → ∞ is concerned, we remark that the condition c = 1, in principle, precludes it. To be precise, if we apply the same scalings (128) to the Lie generators keeping z unchanged and so the product zP 0 as well, then c appears explicitly in the commutators (108), and the r-matrices (131) now read…”

Section: Kinematical Lie Bialgebras and Noncommutative Spacesmentioning

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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“…We recall that the classifications of non-isomorphic 4D and 6D real Drinfel'd double structures were carried out in [120] and [121], respectively, while their Hopf algebra quantizations were constructed in [122]. From these results and also from [50], there were obtained the classifications of Drinfel'd double structures for the (2+1)D (anti-)de Sitter algebras in [111], (2+1)D Poincaré algebra and centrally extended (1+1)D Poincaré algebra in [123], and 3D Euclidean algebra in [124]. By contrast, results concerning Drinfel'd double structures in the (3+1)D case are very scarce, only covering the real so(5) and anti-de Sitter so(3, 2) algebras given in [125].…”

Section: Drinfel'd Double Structuresmentioning

confidence: 99%

Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications

Gutierrez-Sagredo,

Herranz

2021

Preprint

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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 to deal with real structures, it comes out that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we find 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) and for some of their contractions. These geometric results are exhaustively applied onto the (3+1)D kinematical algebras, not only considering the usual (3+1)D spacetime but also the 6D space of lines. We establish 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 speed of light c. We finally obtain 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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Drinfel’d double structures for Poincaré and Euclidean groups

Cited by 4 publications

References 29 publications

Quantum D = 3 Euclidean and Poincaré symmetries from contraction limits

Quantum D = 3 Euclidean and Poincaré symmetries from contraction limits

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

Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications

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