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

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

2006-06-06

arxiv.org/abs/math/0606121v2

SIGMA 2 (2006), 092, 29 pages

Mathematics

Quantum Algebra

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Sym

Scientific paper

10.3842/SIGMA.2006.092

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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