This article is cited in 8 papers

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

A. V. Zotov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: We derive a family of $n$th-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.

Keywords: classical integrable system, $R$-matrix Lax representation, duality.

Received: 08.11.2015
Revised: 18.12.2015

DOI: 10.4213/tmf9097

 English version: Theoretical and Mathematical Physics, 2016, 189:2, 1554–1562

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ArXiv: 1511.02468