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Snyder space-time: K-loop and Lie triple system. (English) Zbl 1220.81145

Summary: Different deformations of the Poincaré symmetries have been identified for various non-commutative spaces (e.g. \(\kappa \)-Minkowski, \(\mathfrak{sl}(2, \mathbb R)\), Moyal). We present here the deformation of the Poincaré symmetries related to Snyder space-time. The notions of smooth “K-loop”, a non-associative generalization of abelian Lie groups, and its infinitesimal counterpart given by the Lie triple system are the key objects in the construction.

MSC:

81R60 Noncommutative geometry in quantum theory
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81T75 Noncommutative geometry methods in quantum field theory
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