Department of Mathematics, University of Munich - B. Pareigis
Ever since the publication of the paper "Quantum Groups" by Drinfel'd in the Proceedings of the International Congress of Mathematicians 1986 in Berkeley (Proc. Vol. 1, 798-820 (1987)) worldwide intensive efforts to join mathematical and physical theories on a high modern standard can be observed.
The field of quantum groups with its many applications in physics and other fields in theoretical physics such as quantum field theory or quantum gravitation are being investigated with the help of various mathematical methods such as theory of Hopf algebras, representation theory, knot theory, category theory, non commutative geometry. Within this large frame we are investigating predominantly algebraic problems in this area.
During the past years detailed solutions for the reconstruction of (quantum) groups, Hopf algebras, and Lie algebras from their categories of representations have been worked out. Furthermore geometrical problems have been studied in connection with non commutative differential calculus. Some of our research activities are being aimed at problems connected with the unusual braidings occurring in connection with quantum groups. They form the basis for a very general definition of Lie algebras whose properties are now being studied.
A further research area is the investigation of finite dynamical systems.
They are being used for (computer) simulations of e.g. large scale traffic
systems, biological systems, finance markets, and electricity networks.
The mathematical theory of these finite dynamical systems is presently
founded and investigated. Mathematical tools from different areas are being
used such as discrete mathematics (graph theory, theory of ordered sets)
and algebra (theory of finite fields, group theory, category theory).