DEFORMATION QUANTIZATION AND NONCOMMUTATIVE GEOMETRY

变形量化和非交换几何

• 批准号: 09640132
• 负责人: YOSHIOKA Akira
• 金额: $ 1.79万
• 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 (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640132/
• 关键词: DEFORMATION QUANTIZATION; STAR PRODUCT; NONCOMMUTATIVE GEOMETRY; ASYMPTOTIC ANALYSIS; SYMPLELTIC MANIFOLD; QUANTIZATION; MECHANICS; Defprmation Quantization; Star Product; シンプレクティック幾何学

We investigate deformation quantization from the view point of Weyl manifolds, representation of algebras, noncommutative Geometry and asymptotic analysis. Main results are the following four points. 1. Noncommutative contact algebra and noncommutative sphere : We intoduce the dass of deformation algebras of noncommutative contact algebras by extending the class of classical commutative algebra from Poisson algebras to contact algebras. We study examples of noncommutative contact algebras such as spheres. 2. Berezin representation of deformation quantization : Using the structure of noncommutative contact algebras, we obtain Berezin representation of deformation quantization. 3. Relation of Weyl manifold and deformation quantization on symplectic manifold : A classification of deformation quantization is given by considering the classification of Weyl manifolds. We obtain the complete invariant of Weyl manifold, which is called Poincare-Cartan invariant. The Poincare-Cartan class is proved to be just the class given by the curvature of Fedosov connection. 4. Asymptotic analysis and partial differential Equations : The basic analysis is studied which is considered important future investigation of deformation quantization. Mainly inverse problem is investigated. The asymptotic distribution of poles of the resolvent is analysed by Nakamura criterion for the Reyleigh wave boundary condition.

我们从Weyl流形、代数表示、非对易几何和渐近分析的角度研究了形变量子化。主要成果有以下四点。1.非对易接触代数与非对易球面：通过将经典的交换代数类从Poisson代数推广到接触代数，我们引入了非对易接触代数的变形代数。我们研究了球面等非对易接触代数的例子。2.形变量子化的Berezin表示：利用非对易接触代数的结构，得到形变量子化的Berezin表示。3.辛流形上的Weyl流形与形变量子化的关系：通过考虑Weyl流形的分类，给出了形变量子化的分类。得到了Weyl流形的完全不变量，称为Poincare-Cartan不变量。证明了Poincare-Cartan类正是由Fedosov联络的曲率给出的类。4.渐近分析和偏微分方程组：对基本分析进行了研究，认为这是变形量子化的重要研究方向。主要研究了反问题。对于Reyleigh波边界条件，用Nakamura准则分析了预解的极点的渐近分布。

项目成果

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki and Akira Yoshioka: "Noncommutative 3-sphere : A model of noncommutative contact algebras" Journal of the mathematical Society of Japan. Vol.50-4. 915-943 (1998)

Hideki Omori、Yoshiaki Maeda、Naoya Miyazaki 和 Akira Yoshioka：“非交换 3 球：非交换接触代数模型”日本数学会杂志。

M.Ikehata, G.Nakamura and M.Yamamoto: "Uniqueness in inverse problems for the isotropic Lame system" J.Math.Sci.Univ.Tokyo. Vol.5. 627-692 (1998)

M.Ikehata、G.Nakamura 和 M.Yamamoto：“各向同性 Lame 系统反演问题的唯一性”J.Math.Sci.Univ.Tokyo。

HIDEKI OMORI: "NONCOMMUTATIVE 3-SPHERE:AMODEL OF NONCOMMUTATIVE CONTACT" JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN. 50・4. 915-943 (1998)

大森秀树：“非交换三球体：非交换接触模型”日本数学会杂志 50・4（1998）。

Hideki Omori: "Noncommutative Contact Algebras" Mathe matical Physics Studies. 20. 333-338 (1997)

Hideki Omori：“非交换接触代数”数学物理研究。

Gen Nakamura: "A Formula For The Fundahental Solution Of Anisotropic Elasticity" Qvarterly J.Appl.Math.50. 179-194 (1997)

Gen Nakamura：“各向异性弹性基本解决方案的公式”Qvarterly J.Appl.Math.50。