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Bicrossproduct structure of \(\kappa\)-Poincaré group and non-commutative geometry. (English) Zbl 1112.81328
Summary: We show that the \(\kappa\)-deformed Poincaré quantum algebra proposed for elementary particle physics has the structure of a Hopf algebra bicrossproduct \(U(\text{so}(1,3))\vartriangleright\!\blacktriangleleft T\). The algebra is a semidirect product of the classical Lorentz group \(\text{so}(1,3)\) acting in a deformed way on the momentum sector \(T\). The novel feature is that the coalgebra is also semidirect, with a backreaction of the momentum sector on the Lorentz rotations. Using this, we show that the \(\kappa\)-Poincaré algebra acts covariantly on a \(\kappa\)-Minkowski space, which we introduce. It turns out necessarily to be deformed and noncommutative. We also connect this algebra with a previous approach to Planck scale physics.

MSC:
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
58B34 Noncommutative geometry (à la Connes)
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