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

This article is cited in 12 scientific papers (total in 12 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.


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

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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
\by A.~N.~Sergeev
\paper Calogero Operator and Lie Superalgebras
\jour TMF
\yr 2002
\vol 131
\issue 3
\pages 355--376
\jour Theoret. and Math. Phys.
\yr 2002
\vol 131
\issue 3
\pages 747--764

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    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
    12. Fedoruk S. Ivanov E. Lechtenfeld O., “Supersymmetric Hyperbolic Calogero-Sutherland Models By Gauging”, Nucl. Phys. B, 944 (2019), 114613  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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