# Trigonometric integrable tops from solutions of associative Yang-Baxter equation

## TL;DR

This paper constructs classical integrable tops from special trigonometric solutions of the associative Yang-Baxter equation, linking quantum R-matrices to classical integrable systems like Calogero-Moser and Ruijsenaars-Schneider models.

## Contribution

It introduces a new class of integrable tops derived from associative Yang-Baxter solutions, detailing their Lax pairs, Poisson structures, and connections to known models.

## Key findings

- Constructed Lax pairs with spectral parameter for these tops.
- Described Poisson structures using linear Poisson-Lie brackets and Sklyanin algebras.
- Established gauge equivalence to well-known integrable models.

## Abstract

We consider a special class of quantum non-dynamical $R$-matrices in the fundamental representation of ${\rm GL}_N$ with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case $N=2$ these are the well-known 6-vertex $R$-matrix and its 7-vertex deformation. The $R$-matrices are used for construction of the classical relativistic integrable tops of the Euler-Arnold type. Namely, we describe the Lax pairs with spectral parameter, the inertia tensors and the Poisson structures. The latter are given by the linear Poisson-Lie brackets for the non-relativistic models, and by the classical Sklyanin type algebras in the relativistic cases. In some particular cases the tops are gauge equivalent to the Calogero-Moser-Sutherland or trigonometric Ruijsenaars-Schneider models.