Lecture Event
Non-standard invariant operators on quaternionic geometries and curved
Casimir operators
Ales Navrat
5 Dec 2007, 11:00 – 12:30
My talk will be devoted to a certain family of non-standard operators on
quaternionic manifolds. Quaternionic structures belong to the wide class
of parabolic geometries and thus a variety of algebraic tools can be
applied in the study of these geometries and of invariant differential
operators. For the homogeneous model, the description of invariant
operators can be reduced to the study of homomorphisms of (generalized)
Verma modules, which are understood in many cases. This concept can be
extended to curved geometries by considering the so-called semi-holonomic
Verma modules, but this does not lead to a complete description. There are
a few operators, which do not arise in this simple algebraic way. Apart
from the critical powers of the Laplace operator on even dimensional
conformal manifolds, the above mentioned invariant operators on
quaternionic geometries provide examples of this situation.
In the end of the lecture, I will describe a new effective approach to
construction of invariant operators, which was recently found by Andreas
Cap and Vladimir Soucek. They proved that the Casimir operator naturally
extends to an invariant differential operator on arbitrary parabolic
geometries and and it can be used to construct invariant operators between
various types of natural bundles.
Weitere Informationen: http://www.mathematik.hu-berlin.de/~baum/
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