Braided geometry is a flavour of noncommutative geometry in which the commutativity is relaxed to some sort of braiding.
This approach has some problems, for example, if the R-matrix is a deformation of the standard flip, then various objects in the category have their graded dimensions different from the classical case, i.e. “the deformation is not flat”. If one replaces the FRT quantized function algebra by reflection equation algebra then, for Hecke R-matrix, the deformation is flat.
Related entries: braid group, quantum Yang-Baxter equation, classical r-matrix, bialgebra cocycle, Drinfeld associator, Drinfeld twist, dynamical Yang-Baxter equation, quasitriangular bialgebra, quasi-Hopf algebra, ribbon category, braiding, braided monoidal category, reflection equation algebra, double Hecke algebra