Multiple Commutation Relations of the Quantum Affine Algebra Uq (glN), Nested Bethe Vector, and the Gelfand–Tsetlin Basis

Abstract We study a certain type of multiple commutation relations of the quantum affine algebra Uq (glN) U q (gl ^ N). We show that all the coefficients in the multiple commutation relations between the L -operator elements are given in terms of the trigonometric weight functions for the vector representation, independent of the representation of the L -operator. For rank one case, our proof also gives a conceptual understanding why the coefficients can also be expressed using the Izergin–Korepin determinants. As a related result, by specializing expressions for the universal nested Bethe vector by Pakuliak–Ragoucy–Slavnov, we also find a construction of the Gelfand–Tsetlin basis for the vector representation using different L -operator elements from the constructions by Nazarov–Tarasov or Molev. We also present corresponding results for the Yangian Yₕ (glN) Y h (gl N).