Noncommutative geometry and quantum symmetry

【摘要】moscovici本项目的目标是完善和进一步发展A. cones和PI最近共同研究的非交换几何中处理量子对称的新见解和新工具，以扩大其应用范围到横向微分几何、量子群的代数和解析理论以及量子场论。主要的新见解，即迪夫等变几何中的对称性是以特定的霍普夫代数的形式组织起来的，将在所有几何上有趣的横向几何类中进行研究，然后用于解决相应的横向指标问题。将循环上同调应用于Hopf代数提供了一个新的工具，用于获得量子群及其作用的新的上同调不变量。提出的工作的动机是开发适当的数学工具来处理非经典空间的大量例子的必要性，这些例子出现在数学和物理的许多领域，其共同特征是它们的可测量局部参数的行为像无限维矩阵而不是数字。这需要对现有几何技术进行相当大的扩展和重新思考，并导致非计算几何的发展。本项目的重点是在存在非经典对称性的情况下对这种非交换空间的处理。

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.