Prof. Dr. Karl-Hermann Neeb
Structure and representation theory of infinite-dimensional Lie groups and Lie algebras
Lie groups and Lie algebras
The simplest examples of Lie groups are are groups of matrices that students already encounter in the basic courses on Linear Algebra. One research focus of the group are infinite dimensional Lie groups, which are generalizations of matrix groups. These are groups carrying a smooth manifold structure for which the group operations are smooth. A natural first order approximation of a Lie group is its Lie algebra and the translation mechanism between Lie groups and Lie algebra lies at the heart of Lie theory. This is analogous to modeling a dynamical system by a differential equation (infinitesimal level) and to recover the dynamics (global level) by an integration process.
The representation theory of Lie groups and their Lie algebras deals with the different realizations of a Lie group by symmetries of a mathematical structure and plays an important role in classifying objects in terms of their symmetries. Felix Klein’s Erlangen program even defines geometry in terms of the corresponding symmetry groups. Lie groups and their Lie algebras also play a fundamental role as symmetries of classical and quantum mechanical systems. Here classical systems are modeled by Hamiltonian Lie group actions and quantum systems lead to unitary representations of Lie groups, an area which is under active investigation for infinite dimensional groups.