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$...

To any germ X of a complex analytic variety with local ring OX one associates the topological Lie algebra Θ(X) = DerOX of vector fields on X. We show that isolated hypersurface singularities X of dimension at least 3 are uniquely determined up to isomorphism by the topological Lie algebra Θ(X).

We show that a connected split reductive group G over a field of characteristic 0 is uniquely determined up to isomorphism by specifying a maximal torus T of G , the set of isomorphism classes of irreducible representations of G , and the character homomorphism from the Grothendieck ring of G to that of T .