Snyder Space-Time: K-Loop and Lie Triple System

Different deformations of the Poincaré symmetries have been identified for various non-commutative spaces (e.g. $\kappa$-Minkowski, $\mathfrak{sl}(2,R)$, 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.