Dept Math & Stat UiT, Forskningsparken B459
  13 Mar 2025 14:00-15:30

Henrik Winther (UiT)

"Higher order connections in noncommutative geometry"

We prove that a system of higher order connections is equivalent to a notion of phase space quantization, in the setting of noncommutative differential geometry. Further we show that higher order connections are equivalent to (ordinary) connections on jet modules. This involves introducing the notion of natural linear differential operator, as well as an important family of examples, namely the Spencer operators, generalizing their corresponding classical analogues.

These Spencer operators form the building blocks of this theory by providing conversions between the different representations of higher order connections as left/right splittings of the jet exact sequences and as explicit pieces of the necessary jet connections which are constructed using the data of a left connection on the first-order differential calculus. A system of such higher order connections then gives a quantization in the sense of a splitting of the projection from differential operator to their symbols. This yields total symbols and star products, i.e. phase space quantizations, but where we allow the position coordinates to form a possibly noncommutative algebra.

Joint work with K. Flood and M. Mantegazza.