# Quantum Trace Formulae for the Integrals of the Hyperbolic Ruijsenaars-Schneider model

## TL;DR

This paper proposes quantum trace formulae for the integrals of motion in the hyperbolic Ruijsenaars-Schneider model, extending classical results through algebraic structures involving the Lax matrix and spectral parameter.

## Contribution

It introduces a conjecture for the quantum analogue of classical trace formulae, connecting classical Poisson reduction with quantum algebraic structures.

## Key findings

- Conjectured quantum trace formulae for the model's integrals of motion.
- Analysis of algebraic structures related to the Lax matrix in classical and quantum contexts.
- Discussion of spectral parameter's role in algebraic structures.

## Abstract

We conjecture the quantum analogue of the classical trace formulae for the integrals of motion of the quantum hyperbolic Ruijsenaars-Schneider model. This is done by departing from the classical construction where the corresponding model is obtained from the Heisenberg double by the Poisson reduction procedure. We also discuss some algebraic structures associated to the Lax matrix in the classical and quantum theory which arise upon introduction of the spectral parameter.