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

TL;DR

This paper explores the deformation of Poincare symmetries in Snyder space-time using the mathematical frameworks of K-loops and Lie triple systems, highlighting their roles in non-commutative geometry.

Contribution

It introduces a novel approach to deform Poincare symmetries in Snyder space-time through K-loops and Lie triple systems, expanding the mathematical tools for non-commutative geometry.

Findings

Deformation of Poincare symmetries in Snyder space-time analyzed.

K-loops and Lie triple systems effectively model non-commutative structures.

Provides a new mathematical framework for non-commutative space-time symmetries.

Abstract

Different deformations of the Poincare symmetries have been identified for various non-commutative spaces (e.g. $\kappa$-Minkowski, $sl(2,R)$, Moyal). We present here the deformation of the Poincare 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.