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Equivalence of Invariant Star-Products: The “Retract” Method

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Geometric Science of Information (GSI 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14071))

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Abstract

In this article, we present a general method for enlarging the group of symmetries (symplectomorphisms) of a given star-product (or deformation quantization) on a symplectic homogeneous space. We call this method the “retract method”.

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Notes

  1. 1.

    This space can be interpreted as the space of flat \({\mathfrak {g}}{}\)-invariant connections on a noncommutative space modelled on the infinite dimensional automorphism group of the star-product (c.f. Vinberg’s description of invariant affine connections on a homogeneous space).

References

  1. Bertelson, M., Bieliavsky, P., Gutt, S.: Parametrizing equivalence classes of invariant star products. Lett. Math. Phys. 46, 339–345 (1998)

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  2. Bieliavsky, P., Detournay, S., Spindel, P.: The deformation quantizations of the hyperbolic plane. Commun. Math. Phys. 289(2), 529–559 (2009)

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  3. Bieliavsky, P., Gayral, V.: Deformation Quantization for Actions of Kählerian Lie groups, vol. 236. Memoirs of the American Mathematical Society (2015)

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  4. Dendoncker, V.: Non-formal Rankin-Cohen deformations. Ph.D. thesis, Université catholique de Louvain (2018)

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  5. Korvers, S.: Quantifications par déformations formelles et non formelles de la boule unité de \(\mathbb{C} ^n\). Ph.D. thesis, Université catholique de Louvain (2014)

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Author information

Authors and Affiliations

  1. Université de Louvain, Ottignies-Louvain-la-Neuve, Belgium

    Pierre Bieliavsky

  2. ICHEC Brussels Management School, Woluwe-Saint-Pierre, Belgium

    Valentin Dendoncker

  3. Gevers Patents, Brussels, Belgium

    Stéphane Korvers

Authors
  1. Pierre Bieliavsky
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  2. Valentin Dendoncker
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  3. Stéphane Korvers
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Corresponding author

Correspondence to Valentin Dendoncker .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

  2. THALES Land and Air Systems, Meudon, France

    Frédéric Barbaresco

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© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

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Bieliavsky, P., Dendoncker, V., Korvers, S. (2023). Equivalence of Invariant Star-Products: The “Retract” Method. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_53

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  • DOI: https://doi.org/10.1007/978-3-031-38271-0_53

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-38270-3

  • Online ISBN: 978-3-031-38271-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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