Noncommutative geometry and quantum symmetry

莫斯科维奇 这个项目的目标是改进和进一步发展新的见解和工具，用于处理非对易几何中的量子力学，这些新的见解和工具来自于A。康纳斯和PI，以扩大其应用范围的横向微分几何，代数和分析理论的量子群，以及量子场论。主要的新见解，即在Diff-equivariantgeometry的对称性来组织在特定的Hopf代数的形式，将被调查的所有几何有趣的类oftransverse几何，然后解决相应的transverse指标问题。主要的新工具，所提供的适应性的循环上同调的Hopf代数，将被应用到获得新的上同调不变量的量子群和他们的作用。提出的工作的动机是必要的developingappropriate的数学工具来处理大量的examplesof非经典空间，出现在数学和物理的许多领域，其共同的特点是，其可测量的局部参数类似于无限维矩阵，而不是数字。这就需要对现有的几何技术进行相当大的扩展和反思，并导致了非对易几何的发展。本项目的重点是在非经典对称性存在的情况下处理这种非对易空间。

AbstractMoscovici The goal of this project is to refine and further develop the new insights and tools for the treatment of quantum symmetryin noncommutative geometry which emerged from the recent joint work of A. Connes and the PI, in order to expand their range of applications to transverse differential geometry, the algebraic and the analytic theory of quantum groups, as well as to quantum field theory. The main new insight, that symmetry in Diff-equivariantgeometry comes organized in the form of specific Hopf algebras,will be investigated for all geometrically interesting classes oftransverse geometries and then employed to settle the correspondingtransverse index problem. The main new tool, provided by the adaptation of cyclic cohomology to Hopf algebras, will be appliedto obtain new cohomological invariants of quantum groups and of theiractions. The proposed work is motivated by the necessity of developingappropriate mathematical tools for handling a wealth of examplesof non-classical spaces, arising in many areas of mathematics and physics, whose common feature is that their measurable local parameters behavelike infinite-dimensional matrices rather than numbers. This requires aconsiderable stretching and rethinking of the existing geometric techniques and led to the development of noncomutative geometry. Theemphasis of the present project is on the the treatment of suchnoncommutative spaces in the presence of non-classical symmetries.