Heisenberg doubles for Snyder type models

TL;DR

This paper explores the application of the Heisenberg double construction to Snyder models, deriving phase space structures and extended algebras, with explicit formulas for different Heisenberg doubles.

Contribution

It introduces a novel application of Heisenberg doubles to Snyder models, including extended algebras and explicit formulae for various constructions.

Findings

Derived the phase space of Snyder models using Heisenberg doubles.

Constructed extended Snyder algebra and group in higher dimensions.

Provided explicit formulas for multiple Heisenberg double realizations.

Abstract

A Snyder model generated by the noncommutative coordinates and Lorentz generators close a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. It leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.