Lie groupoids, Lie algebroids, representations and ideals

Abstract: Symmetries are usually understood as the action of a group. However, many mathematical objects have a structure that we recognise as a symmetry, but which is in fact expressed by a groupoid.

This talk motivates the notions of Lie groupoids and Lie algebroids, and explains why the classical notions of representations and ideals are not sufficient in this context.

Representations up to homotopy of Lie algebroids on graded vector bundles are explained, in particular the adjoint representation up to homotopy of a Lie algebroid is described. The graded trace of the powers of the curvature of a connection up to homotopy induces characteristic classes of graded vector bundles. Using these, one can prove obstructions to the existence of a representation up to homotopy on a graded vector bundle. This gives a further interpretation for the classical Pontryagin characters of vector bundles.

As a consequence, an obstruction to the existence of an infinitesimal ideal system in a Lie algebroid is given -- these objects are considered the right notion of ideal in the context of Lie algebroids.

mathematical physicsalgebraic geometryalgebraic topologyK-theory and homologyoperator algebrasquantum algebrarepresentation theory

Audience: researchers in the topic

Online GAPT Seminar

Series comments: Description: Seminar of the GAPT group at Cardiff University