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

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

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

Bibliographic databases:

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

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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
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
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
4. Langmann E., “Source Identity and Kernel Functions for Elliptic Calogero-Sutherland Type Systems”, Lett Math Phys, 94:1 (2010), 63–75
5. Langmann E., Takemura K., “Source Identity and Kernel Functions for Inozemtsev-Type Systems”, J. Math. Phys., 53:8 (2012), 082105
6. Feigin M., “Generalized Calogero–Moser Systems From Rational Cherednik Algebras”, Sel. Math.-New Ser., 18:1 (2012), 253–281
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
8. Atai F., Langmann E., “Deformed Calogero-Sutherland model and fractional quantum Hall effect”, J. Math. Phys., 58:1 (2017), 011902
9. Sergeev A.N., Veselov A.P., “Symmetric Lie Superalgebras and Deformed Quantum Calogero–Moser Problems”, Adv. Math., 304 (2017), 728–768
10. Farrokh Atai, Edwin Langmann, “Series Solutions of the Non-Stationary Heun Equation”, SIGMA, 14 (2018), 011, 32 pp.
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
12. Fedoruk S. Ivanov E. Lechtenfeld O., “Supersymmetric Hyperbolic Calogero-Sutherland Models By Gauging”, Nucl. Phys. B, 944 (2019), 114613
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