Classification of Six-Dimensional Real Drinfeld Doubles

Šnobl, Libor; Hlavatý, Ladislav

doi:10.1142/s0217751x02010571

Cited by 71 publications

(130 citation statements)

References 9 publications

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“…and coincides with the family of r-matrices in [51,52] up to the change of basis in (62). Note also that the parameter b 2 can be taken to vanish without loss of generality due to the classification of r-matrices for sl(2, R) discussed in the previous section.…”

Section: Lorentzian 3d Constant Curvature Spacetimes As Poisson Homogmentioning

confidence: 79%

“…It follows from the classification of 3d Lie bialgebras and 6d classical doubles in [61,62] that the isometry groups G Λ of constant curvature spacetimes in 3d gravity, which are the groups PSL(2, C), SO(2, 2) and ISO(2, 1) for, respectively, Λ > 0, Λ < 0 and Λ = 0 can all be realised as doubles of certain Poisson-Lie structures on SL(2, R).…”

Section: Coisotropic Lagrangian Subalgebras Of the Double Of A Doublementioning

confidence: 99%

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AdS Poisson homogeneous spaces and Drinfel’d doubles

Ballesteros¹,

Meusburger²,

Naranjo³

2017

J. Phys. A: Math. Theor.

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The correspondence between Poisson homogeneous spaces over a Poisson-Lie group G and Lagrangian Lie subalgebras of the classical double D(g) is revisited and explored in detail for the case in which g = D(a) is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group SL(2, R) ∼ = SO(2, 1), namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on sl(2, R) and as a coisotropic one for the others.We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space AdS 3 and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for so(2, 2), while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on AdS 3 that arise from two Drinfel'd double structures on SO(2, 2). The first one realises AdS 3 as a quotient of SO(2, 2) by the Poisson-subgroup SL(2, R), while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises AdS 3 as a coisotropic Poisson homogeneous space.

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Section: Lorentzian 3d Constant Curvature Spacetimes As Poisson Homogmentioning

confidence: 79%

Section: Coisotropic Lagrangian Subalgebras Of the Double Of A Doublementioning

confidence: 99%

AdS Poisson homogeneous spaces and Drinfel’d doubles

Ballesteros¹,

Meusburger²,

Naranjo³

2017

J. Phys. A: Math. Theor.

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“…In [31] all 6-dimensional Drinfeld doubles were found. Particular care was taken to identify different decompositions into bi-algebras.…”

Section: All Conformal Three Dimensional Dualsmentioning

confidence: 99%

“…Recently all six dimensional Drinfeld doubles have been classified [31,32]. This provides a large class of examples on which Poisson-Lie T-duality can be tested.…”

Section: Introductionmentioning

confidence: 99%

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Poisson-Lie T-plurality

Unge

2002

J. High Energy Phys.

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Abstract:We extend the path-integral formalism for Poisson-Lie T-duality to include the case of Drinfeld doubles which can be decomposed into bi-algebras in more than one way. We give the correct shift of the dilaton, correcting a mistake in the litterature. We then use the fact that the six dimensional Drinfeld doubles have been classified to write down all possible conformal Poisson-Lie T-duals of three dimensional space times and we explicitly work out two duals to the constant dilaton and zero anti-symmetric tensor Bianchi type V space time and show that they satisfy the string equations of motion. This space-time was previously thought to have no duals because of the tracefulness of the structure constants.

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Effective Actions for σ‐Models of Poisson–Lie Type

Jurčo

Vysoký

2019

Fortschritte der Physik

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(Quasi‐)Poisson–Lie T‐duality of string effective actions is described in the framework of generalized geometry of Courant algebroids. The approach is based on a generalization of Riemannian geometry in the context of Courant algebroids, including a proper version of a Levi‐Civita connection. In our approach, the dilaton field is encoded in a Levi‐Civita connection and its form is determined by the Courant algebroid geometry. Explicit examples of background solutions are provided using the approach developed in the paper.

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Classification of Six-Dimensional Real Drinfeld Doubles

Cited by 71 publications

References 9 publications

AdS Poisson homogeneous spaces and Drinfel’d doubles

AdS Poisson homogeneous spaces and Drinfel’d doubles

Poisson-Lie T-plurality

Effective Actions for σ‐Models of Poisson–Lie Type

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