Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 092, 29 pages      math.QA/0606121
Contribution to the Vadim Kuznetsov Memorial Issue

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

Mark J. Hopkins and Alexander I. Molev
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

Received October 14, 2006; Published online December 26, 2006

We prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra Uq(gln). 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 Uq(gln) and show that this limit algebra contains the q-Yangian as a subalgebra.

Key words: quantum affine algebra; quantum Sylvester theorem; skew representations.

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