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

在所谓的“代数群”中，涉及到一些类型的对象。最普遍的是交换代数范畴上的仿射群(方案)或可表示群-函子，它们必然由交换Hopf代数表示。线性代数群就是这样一个仿射群的域中有理点的群，其代表的Hopf代数应该是有限生成的。通过用广义对象、仿射群或Hopf代数代替线性代数群，我们通常可以去掉有限生成、零特征或代数闭性等假设。通过推广到非交换Hopf代数，我们得到了量子群的概念。我们的团队从非交换代数和几何的角度研究了仿射群、超仿射群和Hopf-Galois扩张(或非交换主齐性空间)。

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.