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

一个群G的商由G/N给出，对于G的某个正规子群N;这个对于普通群的平凡事实对于仿射群概型从来不明显。仿射群概型与交换Hopf代数是范畴一一对应的，因此非交换Hopf代数可以看作仿射群概型的量子化对象。首席研究员调查了非交换的Hopf代数的商，称这样的调查'商-Hopf理论'。本研究计划由科学资助基金（C）（2）资助，目的是将仿射-Hopf理论推广到超向量空间的对称范畴或更一般的辫范畴，并将结果应用于微分-差伽罗瓦理论、超仿射群和辫Hopf代数。论文"Picard-Vessiot扩展的artinian简单的模块代数"联合K.天野给出了一个一般框架，以统一伽罗瓦理论的微分方程和差分方程。文章“超仿射群和超形式群中的基本对应”证明了超仿射群和超形式群的基本对应定理。与T. Oka联合发表的论文《单能代数仿射超群和幂零李超代数》超越了众所周知的单能代数仿射群和有限维幂零李代数之间的范畴等价性。最新的预印本在辫子Hopf代数的框架下进一步推广了这一结果.

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.