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 TMF, 2000, Volume 123, Number 2, Pages 205–236 (Mi tmf599)

Orthogonal polynomials of a discrete variable and Lie algebras of complex-size matrices

D. A. Leitesa, A. N. Sergeevb

a Stockholm University
b Balakovo Institute of Technique, Technology and Control

Abstract: We give a uniform interpretation of the classical continuous Chebyshev and Hahn orthogonal polynomials of a discrete variable in terms of the Feigin Lie algebra $\mathfrak{gl}(\lambda)$ for $\lambda\in\mathbb C$. The Chebyshev and Hahn $q$-polynomials admit a similar interpretation, and orthogonal polynomials corresponding to Lie superalgebras can be introduced. We also describe quasi-finite modules over $\mathfrak{gl}(\lambda)$, real forms of this algebra, and the unitarity conditions for quasi-finite modules. Analogues of tensors over $\mathfrak{gl}(\lambda)$ are also introduced.

DOI: https://doi.org/10.4213/tmf599

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English version:
Theoretical and Mathematical Physics, 2000, 123:2, 582–608

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Citation: D. A. Leites, A. N. Sergeev, “Orthogonal polynomials of a discrete variable and Lie algebras of complex-size matrices”, TMF, 123:2 (2000), 205–236; Theoret. and Math. Phys., 123:2 (2000), 582–608

Citation in format AMSBIB
\Bibitem{LeiSer00} \by D.~A.~Leites, A.~N.~Sergeev \paper Orthogonal polynomials of a discrete variable and Lie algebras of complex-size matrices \jour TMF \yr 2000 \vol 123 \issue 2 \pages 205--236 \mathnet{http://mi.mathnet.ru/tmf599} \crossref{https://doi.org/10.4213/tmf599} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1794157} \zmath{https://zbmath.org/?q=an:1017.17021} \elib{https://elibrary.ru/item.asp?id=13340347} \transl \jour Theoret. and Math. Phys. \yr 2000 \vol 123 \issue 2 \pages 582--608 \crossref{https://doi.org/10.1007/BF02551394} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000165897000005} 

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This publication is cited in the following articles:
1. Grozman, P, “The Shapovalov determinant for the Poisson superalgebras”, Journal of Nonlinear Mathematical Physics, 8:2 (2001), 220
2. Sergeev, A, “Enveloping superalgebra U(osp (1 vertical bar 2)) and orthogonal polynomials in discrete indeterminate”, Journal of Nonlinear Mathematical Physics, 8:2 (2001), 229
3. Shreshevskii, IA, “Orthogonalization of graded sets of vectors”, Journal of Nonlinear Mathematical Physics, 8:1 (2001), 54
4. Sergeev A., “Enveloping algebra of GL(3) and orthogonal polynomials”, Noncommutative Structures in Mathematics and Physics, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 22, 2001, 113–124
5. Gargoubi, H, “Algebra gl(lambda) inside the algebra of differential operators on the real line”, Journal of Nonlinear Mathematical Physics, 9:3 (2002), 248
6. Palev, TD, “Jacobson generators, Fock representations and statistics of sl(n+1)”, Journal of Mathematical Physics, 43:7 (2002), 3850
7. Ch. Sachse, “Sylvester–'t Hooft generators and relations between them for $\mathfrak{sl}(n)$ and $\mathfrak{gl}(n|n)$”, Theoret. and Math. Phys., 149:1 (2006), 1299–1311
8. A. V. Lebedev, “On the Bott–Borel–Weil Theorem”, Math. Notes, 81:3 (2007), 417–421
9. Bouarroudj S., Grozman P., Lebedev A., Leites D., “Divided Power (co)homology. Presentations of Simple Finite Dimensional Modular Lie Superalgebras with Cartan Matrix”, Homology Homotopy Appl, 12:1 (2010), 237–278
10. Bouarroudj S., Grozman P., Leites D., “Defining Relations of Almost Affine (Hyperbolic) Lie Superalgebras”, J Nonlinear Math Phys, 17, Suppl. 1 (2010), 163–168
11. Bouarroudj S. Krutov A. Leites D. Shchepochkina I., “Non-Degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras”, Algebr. Represent. Theory, 21:5 (2018), 897–941
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