Theoretical and Mathematical Physics

, Volume 189, Issue 2, pp 1554–1562 | Cite as

Higher-order analogues of the unitarity condition for quantum R-matrices

  • A. V. ZotovEmail author


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.


classical integrable system R-matrix Lax representation duality 


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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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