Integrating Gauss-Manin connection by group cohomology
Makoto Yamashita (Ochanomizu University, Japan)
Tuesday 19th September, 2017 16:00-17:00 Seminar Room 311B
Understanding the linear forms on K-groups given by traces, and more generally by cyclic cocycles, is one of the central questions in operator algebra and noncommutative geometry. When there is a continuous family of algebra structures, the Gauss-Manin connection on periodic cyclic (co)homology due to Getzler, Tsygan, and others in deformation quantization gives a powerful guiding principle on this problem, which is however difficult to substantiate in the operator algebraic (or "strict" deformation) setting. In this talk we consider algebras graded over discrete groups, and consider deformation of product structure by group 2-cocycles. This setting allows us to integrate the Gauss-Manin connection using the natural action of group cohomology, and it recovers previous computations for noncommutative tori and other twisted group algebras. Based partly on joint work with S. Chakraborty (Muenster).