Determinants in quantum matrix algebras and integrable systems

We define quantum determinants in matrix algebras related to pairs of compatible braidings. establish relations between these and the so-called column row determinants, which are often used theory integrable systems. also generalize spin systems using generalized Yangians demonstrate that such not uniquely determined by “quantum coordinate ring” basic space $$V$$ . For example, plane” $$xy=qyx$$ yields two different systems: rational trigonometric.