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Research on Noncommutative Geometry by deformation quantization

基于变形量化的非交换几何研究

基本信息

批准号：
17540096
负责人：
YOSHIOKA Akira
金额：
$ 1.73万
依托单位：
Tokyo University of Science
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

项目摘要

By means differential equations, we extend the Moyal product to exponential functions for quadratic functions with coefficients in complex numbers. With respect to the product, we obtain certain anomalous phenomena such as associativity breaking, doubled valuedness of star exponential functions. Further we define a star product, which is an extension of the Moyal product, by using complex symmetric matrices. An infinitesimal transformation of expression of star products, we obtain a nonlinear connection on the space of functions. We show that the extended transformation of orderings gives a transformation of Cayley type for complex matrices, and we give the explicit formula of the transformation for functions. By studying the parallel transform of the phase functions and the amplitude functions, we made a model of branching of star exponential functions. A commutative product of Moyal type gives several interesting functions such as elliptic functions in the space of exponential functi … More ons of quadratic forms. We also show that several identities for these special functions are given by the star product and simple relation of exponential functions.We call the extended Moyal products K-ordering products and we give a geometrical description of these products: we have an algebraic bundle over the space of all nxn complex symmetric matrices, whose fibers consist of Weyl algebras. Intertwines among K-ordering expressions give a flat connection of this bundle. Flat sections of the bundle are regarded as elements of the Weyl algebra. In this framework, we consider several transcendental elements such as star exponential functions. By means of the star exponential functions of linear forms we can define noncommutative Fourier transform and Laplace transform. By these transforms, we consider noncommutative version of elementary functions, e.g., Gamma functions, delta functions. We also study star exponential functions of quadratic forms on this bundle. These exponential functions give a grebe over the base space and the grebe is described by the bundle and the connection. Less

借助于微分方程组，我们将MoYAL积推广到复数系数二次函数的指数函数。对于乘积，我们得到了某些反常现象，如星指数函数的结合性破缺、重值等。进一步，我们利用复对称矩阵定义了一个星积，它是MoYAL积的推广。通过对星积表达式的一个无穷小变换，我们得到了函数空间上的一个非线性联系。我们证明了序的扩展变换给出了复矩阵的Cayley型变换，并给出了函数变换的显式公式。通过研究位相函数和振幅函数的并行变换，建立了星指数函数的分枝模型。一个Moyal型交换积给出了指数函数…空间中的几个有趣的函数，如椭圆函数更多的二次型。我们还证明了这些特殊函数的几个恒等式是由星积和指数函数的简单关系给出的，我们称这些推广的MoYal积为K-序乘积，并给出了这些乘积的几何描述：我们在所有n×n复对称矩阵的空间上有一个代数丛，它的纤维由Weyl代数组成。K-序表达式之间的缠绕给出了这个丛的平坦连接。丛的平截面被认为是Weyl代数的元素。在这个框架中，我们考虑了几个超越元素，如星指数函数。借助于线性形式的星指数函数，我们可以定义非对易的傅里叶变换和拉普拉斯变换。通过这些变换，我们考虑了初等函数的非对易形式，如Gamma函数、Delta函数。我们还研究了该丛上二次型的星指数函数。这些指数函数给出了基空间上的GREBE，并且GREBE由丛和联络来描述。较少

项目成果

期刊论文数量（31）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Toward Geometric Quantum theory

走向几何量子理论

DOI：
发表时间：
2007
期刊：
Progress in mathematics 252
影响因子：
0
作者：
Hideki Omori;Yoshiaki Maeda;Naoya Miyazaki;Akira Yoshioka;Naoya Miyazaki;Hideki Omori
通讯作者：
Hideki Omori

On the integrability of deformation quantized Toda lattice.

关于形变量子化Toda晶格的可积性。

DOI：
发表时间：
2006
期刊：
Acta Appl. Math. 92
影响因子：
0
作者：
Y.Maeda;H.Omori;et al.;N.Miyazaki
通讯作者：
N.Miyazaki

A splitting theorem for proper complex equivocal submanifolds

真复模歧子流形的分裂定理

DOI：
发表时间：
2006
期刊：
Tohoku Mathematical Journal 58
影响因子：
0
作者：
Yoshiaki Maeda;Peter Michor;Takushiro Ochiai;Akira Yoshioka;Naoya Miyazaki;Naoyuki Koike;Naoyuki Koike;Naoya Miyazaki;Naoyuki Koike
通讯作者：
Naoyuki Koike

Geometric objects in an approach to quantum geometry

量子几何方法中的几何对象