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A First Taste of Quantum Gravity Effects: Deforming Momentum with the Heisenberg Double

Abstract

We present a well-defined framework to deform the phase spaces ofÌýÌýclassical particles. These new phase spaces, called HeisenbergÌýdoubles, provide a laboratory to probe the effects of quantum gravity.ÌýÌý

In particular, they allow us to equip momentum space with aÌýnon-abelian group structure by the introduction of a single deformation parameter. In order to connect Heisenberg doubles withÌýclassical phase spaces we begin with a review of Hamiltonian systems,Ìýsymmetries and conservation laws in the classical framework. Next, weÌýprovide a comprehensive review of the theory behind Poisson-LieÌýgroups, including Lie bialgebras and the construction of the DrinfeldÌýdouble. Lastly, we build the Heisenberg double from Poisson-Lie groupÌýcomponents. We then identify the Heisenberg double as a deformationÌýof the cotangent bundle of Lie groups and extend many of the notionsÌýof classical Hamiltonian systems to this new picture with Poisson-LieÌýsymmetries. As an example, we look at a new presentation of theÌýdeformed rotator.