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

$q$-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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ArXiv: math.QA/0606121
MSC: 81R10
Received: October 14, 2006; Published online December 26, 2006
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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
\Bibitem{HopMol06} \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 \mathnet{http://mi.mathnet.ru/sigma120} \crossref{https://doi.org/10.3842/SIGMA.2006.092} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2280320} \zmath{https://zbmath.org/?q=an:1138.81028} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000207065100091} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84889234765} 

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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
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
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
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)
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
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
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
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