Abstract
A Snyder model generated by the noncommutative coordinates and Lorentz generators closes a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. This 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.
Highlights
Noncommutative coordinates and noncommutative spacetimes lead to a modification of their corresponding relativistic symmetries, which are described by quantum groups (Hopf algebras)
To discuss the extended phase space associated with this extended Snyder model (19) as a result of the Heisenberg double construction, we need to first recall a few facts about the generalized Heisenberg algebra and Weyl realization of the Lorentz algebra based on results presented in [29]
To discuss the phase space corresponding to the extended Snyder space, we consider the Hopf algebra generated by pμν (satisfying (21)), as a dual Hopf algebra B∗, which is equipped with the Hopf algebra structure introduced in [21](see Equation (25) therein), where the coproducts, calculated up to the third order, have the following form:
Summary
Noncommutative coordinates and noncommutative spacetimes lead to a modification of their corresponding relativistic symmetries, which are described by quantum groups (Hopf algebras). In the second part of the paper, to overcome the obstacles raised by non-coassociativity and to construct the full dual Hopf algebra and the full Heisenberg double for the Snyder model, we propose using the extended noncommutative Snyder coordinates . We are able to construct two full Heisenberg doubles, firstly for the extended Snyder algebra generated by tensorial coordinates xμν with its dual Hopf algebra generated by tensorial (conjugate) momenta pρσ. This way, we find the Heisenberg double for the extended Snyder space, which may be considered as the extended Snyder phase space.
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