Canonical and Lie-algebraic twist deformations of Carroll, para-Galilei and Static Hopf algebras

Daszkiewicz, Marcin

doi:10.1142/s0217732319501815

Cited by 7 publications

(8 citation statements)

References 29 publications

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“…where R ′4 = J 04 , J 14 , J 24 , J 34 and R 3 = J 01 , J 02 , J 03 ; note that R ′4 ≃ R 4 and R ′3 ≃ R 3 via D (38). As particular algebras we get the (3+1)D Carroll algebra ii ′ so(3) ≃ i ′ iso(3) formerly introduced in [16,141] (see also [142][143][144][145][146][147] and references therein) and ii ′ so(2, 1) ≃ i ′ iso(2, 1).…”

Section: Drinfel'd Double Structurementioning

confidence: 84%

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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Section: Drinfel'd Double Structurementioning

confidence: 84%

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

Gutierrez-Sagredo,

Herranz

2021

Preprint

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“…In this respect, we remark that, in [145], the (3 + 1)D kinematical algebras were constructed from the static algebra through deformation theory (see [155] for higher dimensions). We also recall that twist deformations for the para-Galilei, static and Carroll algebras were obtained in [146].…”

Section: The Two Remaining Kinematical Algebrasmentioning

confidence: 89%

“…where R 4 = J 04 , J 14 , J 24 , J 34 and R 3 = J 01 , J 02 , J 03 ; note that R 4 R 4 and R 3 R 3 via D (44). As particular algebras, we obtain the (3 + 1)D Carroll algebra ii so(3) i iso(3) formerly introduced in [14,141] (see also [142][143][144][145][146][147] and the references therein) and ii so(2, 1) i iso(2, 1).…”

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confidence: 89%

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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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“…corresponding to the family of three Carrollian algebras c Λ . This comprises the para-Euclidean c + ≃ iso(4) for Λ > 0, the para-Poincaré c − ≃ iso(3, 1) for Λ < 0, and the (proper) Carroll algebra c 0 for Λ = 0 [24,54,[58][59][60][61][63][64][65][66][67][68][69].…”

Section: Ultra-relativistic Limitmentioning

confidence: 99%

Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups

Ballesteros,

Gutierrez-Sagredo,

Herranz

2021

Preprint

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The complete classification of classical r-matrices generating quantum deformations of the (3+1)-dimensional (A)dS and Poincaré groups such that their Lorentz sector is a quantum subgroup is presented. It is found that there exists three classes of such r-matrices, one of them being a novel two-parametric one. The (A)dS and Minkowskian Poisson homogeneous spaces corresponding to these three deformations are explicitly constructed in both local and ambient coordinates. Their quantization is performed, thus giving rise to the associated noncommutative spacetimes, that in the Minkowski case are naturally expressed in terms of quantum null-plane coordinates, and they are always defined by homogeneous quadratic algebras. Finally, non-relativistic and ultra-relativistic limits giving rise to novel Newtonian and Carrollian noncommutative spacetimes are also presented.

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Canonical and Lie-algebraic twist deformations of Carroll, para-Galilei and Static Hopf algebras

Cited by 7 publications

References 29 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

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

Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups

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