D-modules on Noncommutative Spaces. Noncommutative Local Algebra and Representations. Noncommutative Smooth Spaces

本课题主要研究非交换几何：a)非交换局部仿射空间和格式上的d模和微分算子理论。b)非交换局部代数及其在量子包络代数和数学物理相关代数表示理论中的应用。c)非交换光滑局部仿射空间及其相关结构。项目主要目的：研究非交换空间上的d模，特别是量子标志变体和非交换光滑空间上的d模。在非交换空间和格式的情况下，发展d模的结晶方法的模拟。结合d模理论和非交换局部代数的方法，研究量子包络代数和数学物理中其他重要代数的谱和表示。研究非交换光滑空间的性质、重要实例及其应用。研究结果将对非交换几何、非交换代数、表示理论、变形理论和其他一些数学物理主题产生影响。非交换几何是一个相对较新的数学领域，现在正成为数学和理论物理许多领域的重要工具之一，或者说是思维方式。它植根于量子力学和表征理论。但主要的动机来自最近数学物理（量子群和相关的量子化“空间”）和物理学的惊人发展。在最近提出的m理论（现在被认为是描述自然界中存在的所有相互作用的理论的候选理论）中，物理时空的几何是非对易的。该项目的相当一部分，即与光滑非交换空间有关的部分（研究者与M. Kontsevich一起研究），自然与弯曲空间的m理论有关。研究者和他的同事们期望非交换空间的语言和新的直觉不仅将用于“纯粹”数学，而且还将用于现代物理理论。

The project is devoted to the following topics of noncommutative geometry: a) Theory of D-modules and differential operators on noncommutative locally affine spaces and schemes. b) Noncommutative local algebra and its applications to representation theory of quantum enveloping algebras and related algebras of mathematical physics. c) Noncommutative smooth locally affine spaces and related structures. Main purposes of the project: Studying D-modules on noncommutative spaces, in particular on quantum flag varieties and noncommutative smooth spaces. Developing an analogue of the crystallin approach to D-modules in the case of noncommutative spaces and schemes. Combining methods of D-module theory and noncommutative local algebra, study the spectrum and representations of quantum enveloping algebras and other important algebras of mathematical physics. Studying properties and important examples of noncommutative smooth spaces and their applications. Results should have impacts on noncommutative geometry, noncommutative algebra,representation theory, deformation theory and some other topics ofmathematical physics.Noncommutative geometry is a relatively new field of mathematics which is now becoming one of important tools, or ruther ways of thinking, in many areas of mathematics and theoretical physics. It takes roots in quantum mechanics and representation theory. But the main motivations come from recent amazing developments in mathematical physics (quantum groups and related quantized 'spaces') and from physics. In the recently proposed M-theory (which is nowadays regarded as a candidate for the theory describing all interactions existing in the Nature), the geometry of physical space-time is noncommutative. A considerable part of the project, the one concerned with smooth noncommutative spaces (which the investigator studies together with M. Kontsevich), is naturally related to M-theory on curved spaces. The inverstigator and his collegues expect that the language and new intuition of noncommutative spaces will be used not only in 'pure' mathematics, but also in modern physical theories.