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Deformation Quantization for Actions of KĂ€hlerian Lie Groups


About this Title

Pierre Bieliavsky and Victor Gayral

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 236, Number 1115
ISBNs: 978-1-4704-1491-7 (print); 978-1-4704-2281-3 (online)
DOI: http://dx.doi.org/10.1090/memo/1115
Published electronically: December 18, 2014
Keywords:Strict deformation quantization, Symmetric spaces, Representation theory of Lie groups, Deformation of $C^*$-algebras, Symplectic Lie groups, Coherent states, Noncommutative harmonic analysis

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Notations and conventions
  • Chapter 2. Oscillatory integrals
  • Chapter 3. Tempered pairs for KĂ€hlerian Lie groups
  • Chapter 4. Non-formal star-products
  • Chapter 5. Deformation of FrĂ©chet algebras
  • Chapter 6. Quantization of polarized symplectic symmetric spaces
  • Chapter 7. Quantization of KĂ€hlerian Lie groups
  • Chapter 8. Deformation of $C^*$-algebras

Abstract


Let be a Lie group admitting a left-invariant negatively curved KÀhlerian structure. Consider a strongly continuous action of on a Fréchet algebra . Denote by the associated Fréchet algebra of smooth vectors for this action. In the Abelian case and isometric, Marc Rieffel proved that Weyl's operator symbol composition formula (the so called Moyal product) yields a deformation through Fréchet algebra structures on . When is a -algebra, every deformed Fréchet algebra admits a compatible pre--structure, hence yielding a deformation theory at the level of -algebras too. In this memoir, we prove both analogous statements for general negatively curved KÀhlerian groups. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geom,etrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calderón-Vaillancourt Theorem. In particular, we give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.

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