Research of non-commutative differential geometry

非交换微分几何研究

• 批准号: 09440041
• 负责人: OMORI Hideki
• 金额: $ 4.61万
• 依托单位: Science University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-09440041/
• 关键词: non commutative geometry; star-product (*-product); non commutative differential geometry; *-積; 水一積; 幾何学的量子化

The field of deformation quatization is one of the most active, intersecting area of mathematics and physics.There are several kernel places in the world, working to making quantum calculus. It was very important and fruitful to discuss with many people who are interested in this field.In this period of research, the following are discovered :The notion of μ regulated algebras can be the most fundamental notion for the quantum calculus, and for the non-commutative differential geometry, even for the case the Plank constant h is viewed as a genuine parameter moving in positive reals.In such system, we found special elements, that play the same role as vacuum in the classical quantum theory.In the case that h is viewed as a positive real parameter, it is crucial to fix the product formula, for the star-exponential functions of quadratic forms.Here we found strange phenomena that quadratic forms with non-vanishing discriminant have two different inverses and therefore the associativity breaks down.By this phenomenon, we are forced to break some symmetry in order to keep associativity. Infact, we have to use SL (2 ; R)-symmetry instead of SL(2 ; C)-symmetry.

变形定性是数学和物理最活跃的交叉领域之一。世界上有几个核心地区，致力于量子微积分的研究。与许多对此领域感兴趣的人进行讨论是非常重要和富有成果的。在这一研究过程中，发现了以下几点：μ调节代数的概念可以是量子微积分的最基本的概念，对于非交换微分几何，即使在这种情况下，普朗克常数h被视为在正实数中运动的真正参数。在这个系统中，我们发现了特殊的元素，它们在经典量子理论中扮演着与真空相同的角色。在将h视为正实参数的情况下，对于二次型星指数函数，确定乘积公式是至关重要的。这里我们发现了一个奇怪的现象，具有非消失判别式的二次型有两个不同的逆，因此结合律就失效了。通过这种现象，我们被迫打破一些对称性，以保持结合性。事实上，我们必须使用SL(2； R)-对称而不是SL(2； C)-对称。

项目成果

大森英樹: "Noncommutative woeld,and its geometrical picture"A.M.S. translation of Sugaku expositions. (2000)

Hideki Omori：“非交换世界及其几何图画”A.M.S. Sugaku 论述的翻译（2000）。

大森英樹: "Groups of quantum volume preserving diffeomorphisms and their Beregin representation"Analysis on infinite dimensional Lie groups (論文集) World Scientific. 337-354 (1998)

大森秀树：“量子体积保持微分同胚及其 Beregin 表示的群”无限维李群分析（论文集）世界科学 337-354（1998）。

H.Omori, T.Kobayashi: "On global hypoellipticity on compact manifolds"Hokkaido Math.J. 28. 613-633 (1999)

H.Omori、T.Kobayashi：“关于紧流形上的全局亚椭圆性”Hokkaido Math.J.

大森英樹: "非可換の世界と,幾何学的描像" 数学. 50・1. 12-28 (1998)

大森秀树：“非交换世界与几何图” 数学50・1。

H.Omori, Y.Maeda, N.Miyazaki and A.Yoshioka: "Groups of quantum volume pre-serving diffeomorphisms and their Berezin representation."in the book Analysis on infinite-dimensional Lie groups and algebras, ed. H.Heyer, J.Marion, World Scientific. 337-354 (19

H.Omori、Y.Maeda、N.Miyazaki 和 A.Yoshioka：“保留微分同胚的量子体积群及其 Berezin 表示。”在《无限维李群和代数分析》一书中，编辑。