Quasitriangular structure and twisting of the 3D bicrossproduct model

Majid, Shahn; Osei, Prince K.

doi:10.1007/jhep01(2018)147

Cited by 17 publications

(13 citation statements)

References 27 publications

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“…Yet another method relying on quantum deformation of the real involution (⋆-involution) has been studied in [17,60,61]. Assuming q real, the quantum twist (5.30) in the real Hopf algebra (U q (sl(2; C)) ⊗ U q −1 (sl(2; C), ∆ 4 ′ , S 4 ′ , ‡) satisfies the condition (see [17], Prop.…”

Section: Jhep11(2017)187mentioning

confidence: 99%

Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic $$ {\mathfrak{o}}^{\star }(4) $$ symmetries

Borowiec¹,

Lukierski²,

Tolstoy³

2017

J. High Energ. Phys.

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We construct firstly the complete list of five quantum deformations of D = 4 complex homogeneous orthogonal Lie algebra o(4; C) ∼ = o(3; C) ⊕ o(3; C), describing quantum rotational symmetries of four-dimensional complex space-time, in particular we provide the corresponding universal quantum R-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of o(4; C): Euclidean o(4), Lorentz o(3, 1), Kleinian o(2, 2) and quaternionic o ⋆ (4). For o(3, 1) we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra o(4; C) we present new results.

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Section: Jhep11(2017)187mentioning

confidence: 99%

Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic $$ {\mathfrak{o}}^{\star }(4) $$ symmetries

Borowiec¹,

Lukierski²,

Tolstoy³

2017

J. High Energ. Phys.

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“…] op up to some technicalities, and then take the limit). The diagonal twist equivalence between this conventional version of 3D quantum gravity with U (su 2 ) quantum spacetime and the one we began with (on top left) was recently shown by P. Osei and the author [30]. More details of the point of view for 3D quantum gravity are in [32].…”

Section: Duality Principle Aka Quantum Born Reciprocitymentioning

confidence: 63%

“…Alternatively, we can send l c → ∞ and this is the model in the centre of the figure encoding three-dimensional quantum gravity without cosmological constant (to see this one should write U q (so 1,3 ) = U q (su 2 ) U q (h 3 ) = U q (su 2 ) C q [SU 2 ] op up to some technicalities, and then take the limit). The diagonal twist equivalence between this conventional version of three-dimensional quantum gravity with U(su 2 ) quantum space-time and the one we began with (on top left) was recently shown by P. Osei and the author [16]. More details of the point of view for three-dimensional quantum gravity are in [15].…”

Section: Example 22 (Drinfeld-majid Centre)mentioning

confidence: 70%

On the emergence of the structure of physics

Majid

2018

Phil. Trans. R. Soc. A.

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We consider Hilbert's problem of the axioms of physics at a qualitative or conceptual level. This is more pressing than ever as we seek to understand how both general relativity and quantum theory could emerge from some deeper theory of quantum gravity, and in this regard I have previously proposed a or as a key structure. Here, I outline some of my recent work around the idea of quantum space-time as motivated by this non-standard philosophy, including a new toy model of gravity on a space-time consisting of four points forming a square.This article is part of the theme issue 'Hilbert's sixth problem'.

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“…It is also known that the two quantum groups are related by a Drinfeld twist if H is factorisable. This was originally introduced in [43] as an algebraic Wick rotation, and recently applied in [44] to relate the bicrossproduct model quantum spacetime [45] and the fuzzy R 3 quantum spacetime by a module algebra twist as well as to find the universal R-matrix from the former model.…”

Section: Jhep09(2021)210mentioning

confidence: 99%

Semidual Kitaev lattice model and tensor network representation

Girelli

Osei

Osumanu

2021

J. High Energ. Phys.

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Kitaev’s lattice models are usually defined as representations of the Drinfeld quantum double D(H) = H ⋈ H*op, as an example of a double cross product quantum group. We propose a new version based instead on M(H) = Hcop ⧑ H as an example of Majid’s bicrossproduct quantum group, related by semidualisation or ‘quantum Born reciprocity’ to D(H). Given a finite-dimensional Hopf algebra H, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop ⧑ H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.

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Quasitriangular structure and twisting of the 3D bicrossproduct model

Abstract: We show that the bicrossproduct model C[SU * 2 ]◮⊳U (su 2 ) quantum Poincaré group in 2+1 dimensions acting on the quantum spacetime [x i , t] = ıλx i is related by a Drinfeld and module-algebra twist to the quantum double U (su 2 )⊲ Show more

Cited by 17 publications

References 27 publications

Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic $$ {\mathfrak{o}}^{\star }(4) $$ symmetries

Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic $$ {\mathfrak{o}}^{\star }(4) $$ symmetries

On the emergence of the structure of physics

Semidual Kitaev lattice model and tensor network representation

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