On the geometry of the quantum poincaré group

Aschieri, Paolo

doi:10.1016/s0920-5632(97)00326-5

Cited by 1 publication

(1 citation statement)

References 24 publications

(33 reference statements)

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“…The so-called κ-deformation [19][20][21][22][23][24] (which will feature prominently in the rest of the paper) is generated by the r-matrix: r = 1 κ K i ∧ P i . Using (9) we obtain…”

Section: The Meaning Of Coalgebraic Structures: Quantum Spacetimesmentioning

confidence: 99%

Physical constraints on quantum deformations of spacetime symmetries

Mercati

Sergola

2018

Nuclear Physics B

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In this work we study the deformations into Lie bialgebras of the three relativistic Lie algebras: de Sitter, Anti-de Sitter and Poincaré, which describe the symmetries of the three maximally symmetric spacetimes. These algebras represent the centrepiece of the kinematics of special relativity (and its analogue in (Anti-)de Sitter spacetime), and provide the simplest framework to build physical models in which inertial observers are equivalent. Such a property can be expected to be preserved by Quantum Gravity, a theory which should build a length/energy scale into the microscopic structure of spacetime. Quantum groups, and their infinitesimal version 'Lie bialgebras', allow to encode such a scale into a noncommutativity of the algebra of functions over the group (and over spacetime, when the group acts on a homogeneous space). In 2+1 dimensions we have evidence that the vacuum state of Quantum Gravity is one such 'noncommutative spacetime' whose symmetries are described by a Lie bialgebra. It is then of great interest to study the possible Lie bialgebra deformations of the relativistic Lie algebras. In this paper, we develop a characterization of such deformations in 2, 3 and 4 spacetime dimensions motivated by physical requirements based on dimensional analysis, on various degrees of 'manifest isotropy' (which implies that certain symmetries, i.e. Lorentz transformations or rotations, are 'more classical'), and on discrete symmetries like P and T. On top of a series of new results in 3 and 4 dimensions, we find a no-go theorem for the Lie bialgebras in 4 dimensions, which singles out the well-known 'κ-deformation' as the only one that depends on the first power of the Planck length, or, alternatively, that possesses 'manifest' spatial isotropy. 1 We use units in which = c = 1 2 Hopf algebras emerged in several other contexts in physics, e.g. (quantum) integrable models [8], perturbative QCD [9] and AdS/CFT [10] to cite some that are closer to the concerns of the present paper.

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“…The so-called κ-deformation [19][20][21][22][23][24] (which will feature prominently in the rest of the paper) is generated by the r-matrix: r = 1 κ K i ∧ P i . Using (9) we obtain…”

Section: The Meaning Of Coalgebraic Structures: Quantum Spacetimesmentioning

confidence: 99%

Physical constraints on quantum deformations of spacetime symmetries

Mercati

Sergola

2018

Nuclear Physics B

View full text Add to dashboard Cite

show abstract

On the geometry of the quantum poincaré group

Cited by 1 publication

References 24 publications

Physical constraints on quantum deformations of spacetime symmetries

Physical constraints on quantum deformations of spacetime symmetries

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