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Teoreticheskaya i Matematicheskaya Fizika, 2002, Volume 131, Number 3, Pages 355–376
DOI: https://doi.org/10.4213/tmf334
(Mi tmf334)
 

This article is cited in 24 scientific papers (total in 24 papers)

Calogero Operator and Lie Superalgebras

A. N. Sergeev

Balakovo Institute of Technique, Technology and Control
References:
Abstract: We construct a supersymmetric analogue of the Calogero operator $\mathcal S\mathcal L$ which depends on the parameter $k$. This analogue is related to the root system of the Lie superalgebra $\mathfrak {gl}(n|m)$. It becomes the standard Calogero operator for $m = 0$ and becomes the operator constructed by Veselov, Chalykh, and Feigin up to changing the variables and the parameter $k$ for $m = 1$. For $k = 1$ and 1/2, the operator $\mathcal S\mathcal L$ is the radial part of the second-order Laplace operator for the symmetric superspaces corresponding to the respective pairs $(\mathfrak {gl}\oplus \mathfrak {gl}, \mathfrak {gl})$, $(\mathfrak {gl},\mathfrak {osp})$. We show that for any m and n, the supersymmetric analogues of the Jack polynomials constructed by Kerov, Okounkov, and Olshanskii are eigenfunctions of the operator $\mathcal S\mathcal L$. For $k = 1$ and 1/2, the supersymmetric analogues of the Jack polynomials coincide with the spherical functions on the above superspaces. We also study the algebraic analogue of the Berezin integral.
Received: 19.12.2001
English version:
Theoretical and Mathematical Physics, 2002, Volume 131, Issue 3, Pages 747–764
DOI: https://doi.org/10.1023/A:1015968505753
Bibliographic databases:
Language: Russian
Citation: A. N. Sergeev, “Calogero Operator and Lie Superalgebras”, TMF, 131:3 (2002), 355–376; Theoret. and Math. Phys., 131:3 (2002), 747–764
Citation in format AMSBIB
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\by A.~N.~Sergeev
\paper Calogero Operator and Lie Superalgebras
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\yr 2002
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\pages 355--376
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\zmath{https://zbmath.org/?q=an:1039.81028}
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\transl
\jour Theoret. and Math. Phys.
\yr 2002
\vol 131
\issue 3
\pages 747--764
\crossref{https://doi.org/10.1023/A:1015968505753}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000176741900001}
Linking options:
  • https://www.mathnet.ru/eng/tmf334
  • https://doi.org/10.4213/tmf334
  • https://www.mathnet.ru/eng/tmf/v131/i3/p355
  • This publication is cited in the following 24 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теоретическая и математическая физика Theoretical and Mathematical Physics
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    References:69
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