Conformal embeddings and quantum graphs with self-fusion

Abstract

After a short description of the notion of quantum subgroups and quantum modules of quantum groups at roots of unity, in the framework of category theory, and a presentation of the known classifications (that we relate with the theory of conformal embeddings), we sketch several general methods of study and illustrate them on the particular example of a quantum subgroup of the non simple Lie group

SU(2) × SU(3) stemming from a conformal embedding of the later, at level (16, 6), into E8. The graph describing its module structure over the corresponding fusion algebra incorporates several known components graphs of type SU(2) and SU(3), and it has self-fusion, but some of its components don’t. Keywords : quantum symmetries; modular invariance; conformal field theories; quantum groupo¨ıds.