WG1: Cartan Geometry and Representation theory

Cartan Geometry

This WG will focus on the interplay between Cartan theory of connection, tractor calculus and the representation theory of Lie (super)algebras, (super)groups.

Group Leader

Andreas Cap

Universität Wien, Austria

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Goals and Tasks:

G1.1: Give a rigorous approach to representation theory in physics via the theory of Harish-Chandra representations.

G1.2: Develop a unified view via BGG operators and tractor calculus on key questions of differential geometry (subriemannian geodesics, holonomy reduction).

T1.1: Develop a theory of (unitary) representations of semisimple Lie (super) groups in connection with their Cartan geometry application (BGG operators and resolutions) and their physical significance (relativistic description of (super)particles).

T1.2: Study the holonomy reduction of the Cartan connection underlying a suitable parabolic geometry with tractor calculus.

T1.3. Realize the interaction between subriemannian and contact structures by means of the normal solution of certain BGG operators and study the geodesics tangent to the distribution.