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

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

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
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    Citing articles on Google Scholar: Russian citations, English citations
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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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    3. Shreshevskii, IA, “Orthogonalization of graded sets of vectors”, Journal of Nonlinear Mathematical Physics, 8:1 (2001), 54  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    6. Palev, TD, “Jacobson generators, Fock representations and statistics of sl(n+1)”, Journal of Mathematical Physics, 43:7 (2002), 3850  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. A. V. Lebedev, “On the Bott–Borel–Weil Theorem”, Math. Notes, 81:3 (2007), 417–421  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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