The Killing equation and its generalisations for Killing tensors are
important for the study of symmetry, structure deformations, and for
yielding integrable solutions of Hamiltion-Jacobi and Laplacian
equations. These Killing equations arise in some differential
complexes known as BGG complexes. Understanding this is one of the key
steps to a new approach to constructing collections of solutions to
these equations that we outline in this talk. The BGG complex
knowledge enables easy access to the PDE prolongation. The interaction
of the prolonged system with an object known as the scale tractor
yields the desired solutions, which then provide first integrals for
equations such as the geodesic equation, other Hamiliton-Jacobi
equations and for Schrodinger operators equations. The overall story
demonstrates a beautiful interaction of mechanics, geometry, and
algebra.
This is joint work with Simon Goodwin, Thomas Leistner, and Jonathan Kress
Rod Gover