Abstract:
The Coxeter ADE graphs appear in a fundamental way in many
branches of mathematics and physics. The theory of quantum subgroups
which we introduced in connection with the noncommutative Galois
problem provides a multiplicative structure on the vertices of the ADE
graphs. While the ADE graphs have few symmetries, they have quantum
symmetries, with e.g. the E6, E7, E8 graphs having 12, 17 and
respectively 32 quantum symmetries. These give raise to the quantum
subgroup structure, with applications in modular theory and boundary
structures in topological quantum field theory, conformal field theory
and string theory. The quantum group symmetry of a physical theory
breaks to a quantum subgroup at the boundary.
The quantum subgroups of SU(2) construct in a very natural way the
simple Lie groups, with their root systems, canonical bases and
representation theory. The quantum subgroups of SU(3) and SU(4), which
live on higher ADE graphs, consist of a few series and very few
exceptionals; their classification solved a challenge posed by
theoretical physicists. They give raise to higher root systems pointing
toward higher analogs of the simple Lie groups, with potential
applications to models for quantum field theory having a physical
number of dimensions. In algebra representation theory, these
structures generalize the mesh relations of the AuslanderReiten quiver.
Maxim,
