Determinants in quantum matrix algebras and integrable systems

We define quantum determinants in quantum matrix algebras related to pairs of compatible braidings. We establish relations between these determinants and the so-called column and row determinants, which are often used in the theory of integrable systems. We also generalize the quantum integrable spin systems using generalized Yangians related to pairs 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 example, the “quantum plane” $xy=qyx$ yields two different integrable systems: rational and trigonometric.