Quantum Subgroups and Canonical Bases

We show that the most natural and canonical way to construct bases in the universal quantum enveloping algebras of simple Lie groups is provided by the quantum subgroups of SU(2), i.e. the subfactors of Jones index less than 4, and their modules, i.e. the commuting squares. Representation theory is also naturally constructed this way.

The quantum subgroups and modules of other Lie groups and their associated constructions will also be discussed.