Quantum subgroups and quantum symmetry

Adrian Ocneanu, Pennsylvania State University

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. Special announcement: After the talk Prof. Ocneanu will perform arias by Mozart and Rossini for those interested.