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Classical and Quantum Physics pp 337–359Cite as

A Simple Model of Double Dynamics on Lie Groups

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Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 229))

Abstract

We study the dynamics of the rigid rotator on the group manifold of SU(2) as an instance of dynamics on Lie groups together with a dual model whose carrier space is the Borel group \(SB(2,\mathbb {C})\), the Lie Poisson dual of SU(2). We thus introduce a parent action on the Drinfel’d double of the above mentioned groups, which describes the dynamics of a system with twice as many degrees of freedom as the two starting partners. Through a gauging procedure of its global symmetries both the rigid rotor and the dual model are recovered.

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Notes

  1. 1.

    Notice however that, in case G is a compact group, such as SU(2), its cotangent bundle is truly diffeomorphic to its Drinfel’d double, while the cotangent bundle of its dual, \(SB(2\mathbb {C})\) for SU(2), is only a deformation of the double.

  2. 2.

    Noncommutative gauge theories usually require that the gauge group be enlarged (see for example [26] for a review). The differential calculus itself may be bigger than in the commutative case (see for example [27] for an occurrence of this phenomenon in three dimensions and [28] for an application to two-dimensional gauge theory.

    In order to cure nonrenormalizability of noncommutative field theories auxiliary terms have to be added to the action functional, which involve auxiliary parameters. This is the case of simple scalar field theories such as the Grosse-Wulkenhaar model [29], or the translation-invariant model [30].

  3. 3.

    We denote with the symbol \(\bowtie \) the Lie algebra structure of \(\mathfrak {d}\) as a sum of two non-Abelian, non commuting subalgebras, each one of them acting on its dual.

  4. 4.

    The issue of Poisson–Lie symmetries for dynamics such as those discussed in this paper is further elucidated and better understood in the context of field theory in a recent paper [12], where a higher dimensional analogue of the IRR is analyzed. We refer to that for details.

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Acknowledgements

P. V. acknowledges support by COST (European Cooperation in Science and Technology) in the framework of COST Action MP1405 QSPACE.

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Authors and Affiliations

  1. Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II and INFN-Sezione di Napoli, Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126, Napoli, Italy

    Patrizia Vitale

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Correspondence to Patrizia Vitale .

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Editors and Affiliations

  1. Università degli Studi di Napoli Federico II, Naples, Italy

    G. Marmo

  2. Institute of Mathematical Sciences (ICMAT), Madrid, Spain

    David Martín de Diego

  3. Universitat Politècnica de Catalunya, Barcelona, Spain

    Miguel Muñoz Lecanda

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Vitale, P. (2019). A Simple Model of Double Dynamics on Lie Groups. In: Marmo, G., Martín de Diego, D., Muñoz Lecanda, M. (eds) Classical and Quantum Physics. Springer Proceedings in Physics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-030-24748-5_19

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