2006 Fiscal Year Final Research Report Summary
Research on Noncommutative Geometry by deformation quantization
| Project/Area Number |
17540096
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Tokyo University of Science |
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Principal Investigator |
YOSHIOKA Akira Tokyo University of Science, Department of Mathematics Faculty of Science, Professor, 理学部, 教授 (40200935)
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| Co-Investigator(Kenkyū-buntansha) |
HARA Tamio Tokyo University of Science, Department of Mathematics Faculty of Engineering, Professor, 工学部, 教授 (10120205)
KOIKE Naoyuki Tokyo University of Science, Department of Mathematics Faculty of Science, Lecturer, 理学部, 講師 (00281410)
MAEDA Yoshiaki Keio University, Department of Mathematics Faculty of Mathematical Science, Professor, 理工学部, 教授 (40101076)
MIYAZAKI Naoya Keio University, Department of Mathematics Faculty of Economics, Assistant Professor, 経済学部, 助教授 (50315826)
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| Project Period (FY) |
2005 – 2006
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| Keywords | star product / deformation quantization / noncommutative geometry / symplectic geometry / Poisson geometry |
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| Research Abstract |
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
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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
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