Classification of Hopf algebras and quantum groups by tensor equivalences

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Our new results are divided roughly into two. First, we have classified some range of coquasi-bialgebras, which generalize Hopf algebras, up to tensor equivalences of their comodule categories ; this was done by constructing related cohomology groups, in a very concrete way with use of braid diagrams. Second, we have obtained a new result on crossed products, and thereby proved in a unified way three famous theorems for affine group schemes ; this was based on the idea that the Hopf cohomology, which is non-abelian and multiplicative, can be approximated, in some sense, by the HochschiId cohomology, which is abelian and additive.The results were annonced at the following conferences.The 46-th Symposium on Algebra, Osaka Univ., July 30-August 2, 2001.Non-commutative Geometry and Quantum Groups, Banach Center (Warsaw), September 17-29, 2001.