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歧管的角度，代数的表示，非共同几何形状和渐近分析研究了变形量化。主要结果是以下四个点。 1。非交互性接触代数和非交通性球体：我们通过扩展来自Poisson代数的经典交换代数类别来构成非交通式接触代数的变形代数的DASS，以接触代数。我们研究了非交通触点代数（例如球体）的示例。 2。变形量化的Berezin表示：使用非交通触点代数的结构，我们获得了变形量化的Berezin表示。 3. Weyl歧管和变形量化在符号流形上的关系：通过考虑Weyl歧管的分类给出了变形量化的分类。我们获得了Weyl歧管的完整不变，这称为庞加尔 - 卡丹不变。事实证明，Poincare-Cartan类仅是Fedosov Connection的曲率所给的类。 4。渐近分析和部分微分方程：研究基本分析，这被认为是对变形量化的重要未来研究。主要研究了反问题。雷伊利波边界条件中的中村标准分析了分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。

大森 英樹: "非可換世界と,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。

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。