Abstract
We derive a family of nth-order identities for quantum R-matrices of the Baxter–Belavin type in the fundamental representation. The set of identities includes the unitarity condition as the simplest case (n = 2). Our study is inspired by the fact that the third-order identity provides commutativity of the Knizhnik–Zamolodchikov–Bernard connections. On the other hand, the same identity yields the R-matrix-valued Lax pairs for classical integrable systems of Calogero type, whose construction uses the interpretation of the quantum R-matrix as a matrix generalization of the Kronecker function. We present a proof of the higher-order scalar identities for the Kronecker functions, which is then naturally generalized to R-matrix identities.
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 189, No. 2, pp. 176–185, November, 2016.
This research was performed at the Steklov Mathematical Institute of Russian Academy of Sciences and was supported by a grant from the Russian Science Foundation (Project No. 14-50-00005).
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Zotov, A.V. Higher-order analogues of the unitarity condition for quantum R-matrices. Theor Math Phys 189, 1554–1562 (2016). https://doi.org/10.1134/S0040577916110027
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DOI: https://doi.org/10.1134/S0040577916110027