Berezin-type operators on the cotangent bundle of a nilpotent group

Abstract

We define and study coherent states, a Berezin–Toeplitz quantization and covariant symbols on the product \(\varXi \,{:}{=}\,{\mathsf {G}}\times \mathfrak {g}^\sharp \) between a connected simply connected nilpotent Lie group and the dual of its Lie algebra. The starting point is a Weyl system codifying the natural canonical commutation relations of the system. The formalism is meant to complement the quantization of the cotangent bundle \(T^\sharp {\mathsf {G}}\cong {\mathsf {G}}\times \mathfrak {g}^\sharp \) by pseudo-differential operators, to which it is connected in an explicit way. Some extensions are indicated, concerning \(\tau \)-quantizations and variable magnetic fields.

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Correspondence to M. Măntoiu.

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M. M. has been supported by the Fondecyt Project No. 1160359.

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Măntoiu, M. Berezin-type operators on the cotangent bundle of a nilpotent group. J. Pseudo-Differ. Oper. Appl. 10, 535–555 (2019). https://doi.org/10.1007/s11868-019-00297-z

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Keywords

  • Nilpotent group
  • Lie algebra
  • Coherent states
  • Pseudo-differential operator
  • Symbol
  • Berezin quantization

Mathematics Subject Classification

  • Primary 22E25
  • 47G30
  • Secondary 22E45
  • 46L65