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\(\ell\)-weights and factorization of transfer operators

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Abstract

We analyze the \(\ell\)-weights of the evaluation and \(q\)-oscillator representations of the quantum loop algebras \(\mathrm U_q(\mathcal L(\mathfrak{sl}_{l+1}))\) for \(l=1\) and \(l=2\) and prove factorization relations for the transfer operators of the associated quantum integrable systems.

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Notes

  1. For the terminology used here, we refer the reader to [14] and Sec. 5 below.

  2. We let \(\mathbb E_{ab}\) denote the usual matrix units.

  3. For the explanation of the notation, see, e.g., [52].

  4. In this paper, we always assume that the representations used are in the category \(\mathcal O\).

  5. An element \(x\in\mathrm U_q(\mathcal L(\mathfrak g))\) is called group-like if \(\Delta(x)=x\otimes x\).

  6. One can also use the tensor product \(\psi_{\eta_1}\otimes_\Delta\dotsb\otimes_\Delta\psi_{\eta_n}\) as \(\psi\). However, we do not consider such a generalization in this paper.

  7. Here and hereafter, we treat any \(\mathrm U_q(\mathcal L(\mathfrak{sl}_{l+1}))\)-module as the corresponding \(\mathrm U_q(\mathcal L(\mathfrak b_+))\)-module.

  8. We set \(\mathbf n'=(n_{21})\). It may seem that we are using unnecessarily cumbersome notation. This is justified by the fact that we use the same notation for all values of \(l\).

  9. We now set \(\mathbf n'=(n_{22},n_{31},n_{32})\).

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Acknowledgments

The author is grateful to H. Boos, F. Göhmann, A. Klümper, and Kh. S. Nirov, in collaboration with whom some important results used in this paper were previously obtained, for the useful discussions.

Funding

This work was supported in part by the RFBR grant no. 20-51-12005.

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  1. Logunov Institute for High Energy Physics, National Research Center “Kurchatov Institute”, Protvino, Moscow region, Russia

    A. V. Razumov

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 208, pp. 310-341 https://doi.org/10.4213/tmf10082.

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Razumov, A.V. \(\ell\)-weights and factorization of transfer operators. Theor Math Phys 208, 1116–1143 (2021). https://doi.org/10.1134/S0040577921080092

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