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SIGMA, 2006, Volume 2, 092, 29 pages (Mi sigma120)  

This article is cited in 7 scientific papers (total in 7 papers)

$q$-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra

Mark J. Hopkins, Alexander I. Molev

School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

Abstract: We prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra $\mathrm U_q(\widehat{\mathfrak{gl}}_n)$. We then use it to give an explicit realization of the skew representations of the quantum affine algebra. This allows one to identify them in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand–Tsetlin character or ($q$-character). We also apply the quantum Sylvester theorem to construct a $q$-analogue of the Olshanski algebra as a projective limit of certain centralizers in $\mathrm U_q(\mathfrak{gl}_n)$ and show that this limit algebra contains the $q$-Yangian as a subalgebra.

Keywords: quantum affine algebra; quantum Sylvester theorem; skew representations

DOI: https://doi.org/10.3842/SIGMA.2006.092

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Full text: http://emis.mi.ras.ru/.../Paper092
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Bibliographic databases:

ArXiv: math.QA/0606121
MSC: 81R10
Received: October 14, 2006; Published online December 26, 2006
Language:

Citation: Mark J. Hopkins, Alexander I. Molev, “A $q$-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra”, SIGMA, 2 (2006), 092, 29 pp.

Citation in format AMSBIB
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\by Mark J.~Hopkins, Alexander I.~Molev
\paper A~$q$-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra
\jour SIGMA
\yr 2006
\vol 2
\papernumber 092
\totalpages 29
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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. Chervov, A, “Algebraic properties of Manin matrices 1”, Advances in Applied Mathematics, 43:3 (2009), 239  crossref  mathscinet  zmath  isi  scopus
    2. Li J., “The quantum Casimir operators of U-q(gl(n)) and their eigenvalues”, Journal of Physics A-Mathematical and Theoretical, 43:34 (2010), 345202  crossref  mathscinet  zmath  isi  scopus
    3. Chen H., Guay N., “Twisted Affine Lie Superalgebra of Type Q and Quantization of its Enveloping Superalgebra”, Math. Z., 272:1-2 (2012), 317–347  crossref  mathscinet  zmath  isi  scopus
    4. Mironov, A., Morozov, A., Runov, B., Zenkevich, Y., Zotov, A., “Spectral dualities in XXZ spin chains and five dimensional gauge theories”, Journal of High Energy Physics, 2013:12 (2013)  crossref  mathscinet  scopus
    5. Tsuboi Z., “Asymptotic Representations and Q-Oscillator Solutions of the Graded Yang–Baxter Equation Related To Baxter Q-Operators”, Nucl. Phys. B, 886 (2014), 1–30  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Frappat L., Jing N., Molev A., Ragoucy E., “Higher Sugawara Operators for the Quantum Affine Algebras of Type A”, Commun. Math. Phys., 345:2 (2016), 631–657  crossref  mathscinet  zmath  isi  elib  scopus
    7. Futorny V., Hartwig J.T., “De Concini-Kac Filtration and Gelfand-Tsetlin Generators For Quantum Gl(N)”, Linear Alg. Appl., 568 (2019), 173–188  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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