# Relativistic elliptic matrix tops and finite Fourier transformations

## TL;DR

This paper explores a family of classical elliptic integrable systems, revealing a symmetry between spectral and relativistic parameters via finite Fourier transformations, and introduces models of interacting matrix tops with flexible parameter roles.

## Contribution

It introduces a novel symmetry between spectral and relativistic parameters in elliptic integrable systems using finite Fourier transforms and describes new models of interacting matrix tops.

## Key findings

- Finite Fourier transformation symmetry between parameters.
- Flexible choice of spectral and relativistic parameters.
- Models of interacting $GL(M)$ and $GL(N)$ matrix tops.

## Abstract

We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the "off-shell" Lax pairs, which do not satisfy the Lax equations in general case but become true Lax pairs under various conditions (reductions). At the level of the off-shell Lax matrix there is a natural symmetry between the spectral parameter $z$ and relativistic parameter $\eta$. It is generated by the finite Fourier transformation, which we describe in detail. The symmetry allows to consider $z$ and $\eta$ on an equal footing. Depending on the type of integrable reduction any of the parameters can be chosen to be the spectral one. Then another one is the relativistic deformation parameter. As a by-product we describe the model of $N^2$ interacting $GL(M)$ matrix tops and/or $M^2$ interacting $GL(N)$ matrix tops depending on a choice of the spectral parameter.