Multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914337995620352 |
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| author | Gerrard, Allan John Motegi, Kohei Sakai, Kazumitsu |
| author_facet | Gerrard, Allan John Motegi, Kohei Sakai, Kazumitsu |
| contents | We study a certain type of multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$. 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 $Y_h(\mathfrak{gl}_N)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21233 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis Gerrard, Allan John Motegi, Kohei Sakai, Kazumitsu Quantum Algebra Mathematical Physics Exactly Solvable and Integrable Systems We study a certain type of multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$. 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 $Y_h(\mathfrak{gl}_N)$. |
| title | Multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis |
| topic | Quantum Algebra Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2510.21233 |