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

对于G的正规子群N，群G的商由G/N给出；对于普通群来说，这个微不足道的事实对于仿射群方案来说永远不明显。仿射群方案与可交换Hopf代数一对一对应，因此非可交换Hopf代数可以看作是仿射群方案的量子化对象。首席研究者研究了非交换Hopf代数的商，称这种研究为“商-Hopf理论”。本课题由科学基金(C)(2)资助，将商-Hopf理论推广到超向量空间的对称范畴或更一般地推广到编织范畴，并将结果应用于微分-差分伽罗瓦理论、超仿射群和编织Hopf代数。与K.Amano合著的“单模代数的Picard-Vessiot扩展”给出了统一微分方程和差分方程的伽罗瓦理论的一般框架。本文“超仿射群和超形式群的基本对应”证明了超仿射群和超形式群的基本对应定理。与T.Oka联合的“单幂代数仿射超群与幂零李超代数”，将众所周知的单幂代数仿射群与有限维幂零李代数之间的范畴等价进行了扩充。最新的预印本将结果进一步推广到编织Hopf代数的框架中。

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.