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

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.

arXiv commentThis is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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• Asymptotic representations and q-oscillator solutions of the graded Yang-Baxter equation related to Baxter Q-operators
• Higher Sugawara operators for the quantum affine algebras of type A
• Non-commutative Sylvester's determinantal identity
• Eigenvalues of quantum Gelfand invariants
• On the primality of totally ordered $q$-factorization graphs
• De Concini-Kac filtration and Gelfand-Tsetlin generators for quantum gl_N
• R-Matrix presentation of quantum affine algebra in type $A_{2n-1}^{(2)}$
• Twisted q-Yangians and Sklyanin determinants