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

Hopf 商理论的推广及其在子因素和其他因素中的应用

基本信息

批准号：
16540008
负责人：
MASUOKA Akira
金额：
$ 1.54万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

项目摘要

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.

对于 G 的某个正规子群 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从来不明显。仿射群方案与交换Hopf代数是一一对应的，因此非交换Hopf代数可以被视为仿射群方案的量化对象。首席研究员研究了非交换霍普夫代数的商，称这种研究为“商霍普夫理论”。该研究项目由科学补助金 (C)(2) 支持，旨在推广商 Hopf 理论以适应超矢量空间的对称范畴或更一般地适应编织范畴，并应用结果微分差分伽罗瓦理论、超仿射群和辫状 Hopf 代数。与 K.Amano 合作的论文“Picard-Vessiot extensions of artinian simple module algebras”给出了统一微分方程和差分方程的伽罗瓦理论的通用框架。 “超仿射群和超形式群中的基本对应关系”一文证明了超仿射群和超形式群的基本对应定理。与 T.Oka 合作的论文“单能代数仿射超群和幂零李超代数”将单能代数仿射群和有限维幂零李代数之间众所周知的范畴等价性进行了超级化。最新的预印本将这一结果进一步推广到辫状 Hopf 代数的框架中。