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

Mark J. Hopkins·Alexander I. Molev
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Abstract

We prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra ${\rm U}_q(\hat{\mathfrak{gl}}_n)$. 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 ${\rm U}_q(\mathfrak{gl}_n)$ and show that this limit algebra contains the $q$-Yangian as a subalgebra.

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BibTeXRIS

Mark J. Hopkins, Alexander I. Molev. 2006-12-26. A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra. https://doi.org/10.3842/sigma.2006.092

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