Noncommutative geometry and quantum symmetry

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

Abstract Moscovici The goal of this project is to refine and further develop the new insights and tools for the treatment of quantum symmetry in 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-equivariant geometry comes organized in the form of specific Hopf algebras, will be investigated for all geometrically interesting classes of transverse geometries and then employed to settle the corresponding transverse index problem. The main new tool, provided by the adaptation of cyclic cohomology to Hopf algebras, will be applied to obtain new cohomological invariants of quantum groups and of their actions. The proposed work is motivated by the necessity of developing appropriate mathematical tools for handling a wealth of examples of non-classical spaces, arising in many areas of mathematics and physics, whose common feature is that their measurable local parameters behave like infinite-dimensional matrices rather than numbers. This requires a considerable stretching and rethinking of the existing geometric techniques and led to the development of noncomutative geometry. The emphasis of the present project is on the the treatment of such noncommutative spaces in the presence of non-classical symmetries.