Classification of coquasi-Hopf algebras by tensor equivalence and constructions of new braidings

In what is called "Algebraic Groups" some kinds of objects are involved. The most generalized are affine group (schemes), or representable group-functors on the category of commutative algebras, which are necessarily represented by commutative Hopf algebras. A linearly algebraic group is precisely the group of rational points in a field of such an affine group whose representative Hopf algebra is supposed to be finitely generated. By replacing linear algebraic groups with the generalized object, affine groups or Hopf algebras, we can often remove assumptions such as finite generation, zero characteristic or algebraic closedness. By extending to non-commutative Hopf algebras, we reach the notion of quantum groups. Our team has investigated affine groups, super-affine groups and Hopf-Galois extensions (or non-commutative principal homogeneous spaces) from the view-point of non-commutative algebra and geometry.