Quantum subgroups are module categories, which encode the ``higher representation theory'' of the Lie algebras. They appear naturally in mathematical physics, where they correspond to extensions of the Wess-Zumino-Witten models. The classification of these quantum subgroups has been a long-standing open problem. The main issue at hand being the possible existence of exceptional examples. Despite considerable attention from both physicists and mathematicians, full results are only known for sl_2 and sl_3.
In this talk I will describe recent progress in the quantum subgroup classification program for sl_n. Our results finish off the classification for n = 5,6,7, and pave the way for higher ranks. In particular we discover several new exceptionals.
Despite being an algebraic question at heart, our techniques draw heavily from the theory of operator algebras. In particular Vaughan Jones planar algebras, and the Cuntz algebras make key appearances.