# Determinants in Quantum Matrix Algebras and Integrable Systems

## TL;DR

This paper introduces quantum determinants in Quantum Matrix Algebras, explores their relation to determinants in integrable systems, and generalizes quantum spin systems using generalized Yangians, revealing non-uniqueness in their quantum coordinate rings.

## Contribution

It defines quantum determinants related to compatible braidings and generalizes quantum integrable spin systems with new insights into their dependence on quantum coordinate rings.

## Key findings

- Quantum determinants relate to column- and row-determinants in integrable systems.
- Generalized Yangians lead to new quantum spin systems.
- Quantum plane xy=qyx yields both rational and trigonometric integrable systems.

## Abstract

We define quantum determinants in Quantum Matrix Algebras, related to couples of compatible braidings following the scheme from [G]. We establish relations between these determinants and the so-called column-(row-)determinants, often used in the theory of integrable systems. Also, we generalize the quantum integrable spin systems from [CFRS] by using generalized Yangians, related to couples of compatible braidings. We demonstrate that such quantum integrable spin systems are not uniquely determined by the "quantum coordinate ring" of the basic space V. For instance, the "quantum plane" xy=qyx gives rise to two different integrable systems: rational and trigonometric ones.