RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



TMF:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


TMF, 2002, Volume 131, Number 3, Pages 355–376 (Mi tmf334)  

This article is cited in 11 scientific papers (total in 11 papers)

Calogero Operator and Lie Superalgebras

A. N. Sergeev

Balakovo Institute of Technique, Technology and Control

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.

DOI: https://doi.org/10.4213/tmf334

Full text: PDF file (346 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2002, 131:3, 747–764

Bibliographic databases:

Received: 19.12.2001

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
\Bibitem{Ser02}
\by A.~N.~Sergeev
\paper Calogero Operator and Lie Superalgebras
\jour TMF
\yr 2002
\vol 131
\issue 3
\pages 355--376
\mathnet{http://mi.mathnet.ru/tmf334}
\crossref{https://doi.org/10.4213/tmf334}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1931149}
\zmath{https://zbmath.org/?q=an:1039.81028}
\elib{http://elibrary.ru/item.asp?id=13395460}
\transl
\jour Theoret. and Math. Phys.
\yr 2002
\vol 131
\issue 3
\pages 747--764
\crossref{https://doi.org/10.1023/A:1015968505753}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000176741900001}


Linking options:
  • http://mi.mathnet.ru/eng/tmf334
  • https://doi.org/10.4213/tmf334
  • http://mi.mathnet.ru/eng/tmf/v131/i3/p355

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Sergeev, A, “Generalised discriminants, deformed Calogero–Moser-Sutherland operators and super-Jack polynomials”, Advances in Mathematics, 192:2 (2005), 341  crossref  mathscinet  zmath  isi  scopus
    2. Macedo-Junior, AF, “Brownian-motion ensembles of random matrix theory: A classification scheme and an integral transform method”, Nuclear Physics B, 752:3 (2006), 439  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Hallnas M., Langmann E., “A Unified Construction of Generalized Classical Polynomials Associated with Operators of Calogero-Sutherland Type”, Constructive Approximation, 31:3 (2010), 309–342  crossref  mathscinet  zmath  isi  elib  scopus
    4. Langmann E., “Source Identity and Kernel Functions for Elliptic Calogero-Sutherland Type Systems”, Lett Math Phys, 94:1 (2010), 63–75  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Langmann E., Takemura K., “Source Identity and Kernel Functions for Inozemtsev-Type Systems”, J. Math. Phys., 53:8 (2012), 082105  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Feigin M., “Generalized Calogero–Moser Systems From Rational Cherednik Algebras”, Sel. Math.-New Ser., 18:1 (2012), 253–281  crossref  mathscinet  zmath  isi  elib  scopus
    7. Atai F. Hallnaes M. Langmann E., “Source Identities and Kernel Functions For Deformed (Quantum) Ruijsenaars Models”, Lett. Math. Phys., 104:7 (2014), 811–835  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. Atai F., Langmann E., “Deformed Calogero-Sutherland model and fractional quantum Hall effect”, J. Math. Phys., 58:1 (2017), 011902  crossref  mathscinet  zmath  isi  scopus
    9. Sergeev A.N., Veselov A.P., “Symmetric Lie Superalgebras and Deformed Quantum Calogero–Moser Problems”, Adv. Math., 304 (2017), 728–768  crossref  mathscinet  zmath  isi  scopus
    10. Farrokh Atai, Edwin Langmann, “Series Solutions of the Non-Stationary Heun Equation”, SIGMA, 14 (2018), 011, 32 pp.  mathnet  crossref
    11. Atai F., Hallnas M., Langmann E., “Orthogonality of Super-Jack Polynomials and a Hilbert Space Interpretation of Deformed Calogero-Moser-Sutherland Operators”, Bull. London Math. Soc., 51:2 (2019), 353–370  crossref  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
    Number of views:
    This page:211
    Full text:92
    References:38
    First page:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019