Twist for Snyder space

TL;DR

This paper constructs a twist operator for Snyder space, revealing a non-associative Hopf algebra structure that could impact the development of field theories on this noncommutative geometry.

Contribution

It provides the first explicit analytical twist operator for Snyder space, linking non-associative star products with Hopf algebroid structures.

Findings

Reproduces correct momentum and Lorentz coproducts

Describes twisted Poincaré symmetry as non-associative Hopf algebra

Identifies undeformed Lorentz symmetry in the twisted framework

Abstract

We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincar\'{e} symmetry is described by a non-associative Hopf algebra, while the twisted Lorentz symmetry is described by the undeformed Hopf algebra. This new twist might be important in the construction of different types of field theories on Snyder space.