Abstract
The quantum duality principle is used to obtain explicitly the Poisson analogue of the kappa-(A)dS quantum algebra in (3+1) dimensions as the corresponding Poisson-Lie structure on the dual solvable Lie group. The construction is fully performed in a kinematical basis and deformed Casimir functions are also explicitly obtained. The cosmological constant $\Lambda$ is included as a Poisson-Lie group contraction parameter, and the limit $\Lambda\to 0$ leads to the well-known kappa-Poincar\'e algebra in the bicrossproduct basis. A twisted version with Drinfel'd double structure of this kappa-(A)dS deformation is sketched.
Highlights
Since Classical Gravity is essentially a theory describing the geometry of spacetime, it seems natural to consider that a suitable definition of a “quantum” spacetime geometry emerging at the Planck energy regime could be a reasonable feature of Quantum Gravity
Quantum groups [6,7,8] provide a consistent approach to noncommutative spacetimes, since the latter are obtained as noncommutative algebras that are covariant under the action of quantum kinematical groups
The wellknown κ -Minkowski spacetime [9,10,11,12] was obtained as a byproduct of the κ -Poincaré quantum algebra, which was introduced in [13] by making use of quantum group contraction techniques [17,18]
Summary
Since Classical Gravity is essentially a theory describing the geometry of spacetime, it seems natural to consider that a suitable definition of a “quantum” spacetime geometry emerging at the Planck energy regime could be a reasonable feature of Quantum Gravity. The explicit expression of the two Casimir functions is obtained and the κ -Poincaré limit is straightforwardly computed This method can be further applied to the twisted κ -AdSω algebra arising from a Drinfel’d double structure in (3+1) dimensions, and whose first-order noncommutative spacetime has been recently introduced in [55]. We stress that we have obtained a two-parametric deformation, which is ruled by a “quantum” deformation parameter z = 1/κ together with a “classical” deformation parameter ω (the cosmological constant) Notice that this coproduct is written in a “bicrossproduct-type” basis that generalizes the one corresponding to the (2+1) κ -AdSω algebra [38].
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