Generalization of Hopf-quotient theory and applications to subfactors and others

A quotient of a group G is given by G/N for some normal sub-group N of G ; this trivial fact for ordinary groups is never obvious for affine group schemes. Affine group schemes are in a categorical one-to-one correspondence with commutative Hopf algebras, and therefore non-commutative Hopf algebras can be regarded as quantized objects of affine group schemes. The head investigator has investigated quotients of non-commutative Hopf algebras, calling such an investigation 'Quotient-Hopf Theory'. This research project supported by the Grant-in-Aid for Scientific (C)(2) is to generalize the quotient-Hopf theory to fit in the symmetric category of super-vectoe spaces or more genrally in braided categories, and to apply the results to differential-difference Galois theories, super affine groups and braided Hopf algebras. The paper "Picard-Vessiot extensions of artinian simple module algebras" joint with K.Amano gives a general framework to unify Galois theories for differential equations and difference equations. The article "The fundamental correspondences in super affine groups and super formal groups" proves the fundamental correspondence theorems for super affine groups and super formal groups, both. The paper "Unipotent algebraic affine supergroups and nilpotent Lie superalgebras" joint with T.Oka superizes the well-known category-equivalence between unipotent algebraic affine groups and finite-dimensional nilpotent Lie algebras. The result was further generalized in the framework of braided Hopf algebras by the newest preprint.