Multiple commutation relations of the quantum affine algebra U_q(\widehat\mathfrakgl_N), nested Bethe vector and the Gelfand-Tsetlin basis

2025/10/24 by Gerrard, Allan John, Motegi, Kohei, Sakai, Kazumitsu
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2510.21233

Abstract

We study a certain type of multiple commutation relations of the quantum affine algebra U_q(\widehat\mathfrakgl_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 Y_h(\mathfrakgl_N).

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