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Symmetries, Integrable Systems and Representations pp 483–490Cite as

Non-commutative Harmonic Oscillators

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 40))

Abstract

This is a survey on the non-commutative harmonic oscillator, which is a generalization of usual (scalar) harmonic oscillators to the system introduced by Parmeggiani and Wakayama. With the definitions and the basic properties, we summarize the positivity of several related operators with \(\mathfrak{sl}_{2}\) interpretations. We also mention some unsolved questions, in order to clarify the current status of the problems and expected further development.

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Acknowledgements

This work is partially supported by JSPS Grant-in-Aid for Scientific Research (A) #19204011, and JST CREST.

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Authors and Affiliations

  1. Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan

    Hiroyuki Ochiai

Authors
  1. Hiroyuki Ochiai
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Correspondence to Hiroyuki Ochiai .

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Editors and Affiliations

  1. Inst. Camille Jordan, UMR5028 CNRS, Université Claude Bernard Lyon 1, Villeurbanne Cedex, 69622, France

    Kenji Iohara

  2. Inst. de Mathém. de Jus., UMR 7586 CNRS, Université Pierre et Marie Curie, Place Jussieu, Case 247 4, Paris, 75252, France

    Sophie Morier-Genoud

  3. Inst. Camille Jordan, UMR5028 CNRS, Université Claude Bernard Lyon I, Villeurbanne Cedex, 69622, France

    Bertrand Rémy

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In honor of Professor Jimbo’s 60th birthday.

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Ochiai, H. (2013). Non-commutative Harmonic Oscillators. In: Iohara, K., Morier-Genoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_19

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