Cyclic homology and quantum group symmetry

Algebraic geometry and global analysis demonstrate that parts of geometry and topology can be reformulated effectively in terms of suitable rings of functions on geometric spaces (manifolds, varieties etc.). Noncommutative geometry aims to go further and to extend the resulting theory purely algebraically towards general noncommutative rings. On one hand this displays the intrinsic setting and generality in which certain concepts and results can be formulated, and on the other hand it led through their application to specific noncommutative rings to connections to subjects ranging from number theory to theoretical physics. Homological techniques play a central role in this theory, and especially in the approach of Alain Connes. His programme is centred around far-reaching generalisations of the Atiyah-Singer index theorem and the involved analogue of the classical Chern character. On the conceptual side, Connes' most influential discovery was probably cyclic homology, a subtle substitute of de Rham theory in the framework of noncommutative geometry.The background of the proposed research project is the attempt to apply these methods to algebras obtained by deformation quantisation. The latter formalises the passage from a mechanical system to its counterpart in quantum mechanics and attaches certain noncommutative algebras to Poisson structures on manifolds or affine varieties. Applying this to Lie groups and algebraic groups yields quantum groups that have found in the last 25 years several applications especially in knot theory and in quantum statistical mechanics. As recent work of several authors indicates, this attempt could lead to substantial generalisations of Connes' well-established machinery. Sufficiently nontrivial Poisson structures give rise to a modular class which is represented on the quantum level by a certain automorphism of the algebra under consideration (see the Case of Support for more details). It became clear that this automorphism can be incorporated at several places into the theory and that this is natural for several reasons, but the overall picture is still unclear.The proposed project will investigate some aspects of this incorporation of modularity into noncommutative geometry, focusing in particular on cyclic homology itself. This seems a natural next step in the development of noncommutative geometry. On the other hand, the applications of the generalised methods to quantum groups could provide new stimulations for example for Woronowicz's theory of covariant differential calculi or for the construction and study of physical models with quantum group symmetry.