Classical Symmetries and Quantum Subgroups
In the early 2000’s Ocneanu initiated the classification of quantum subgroups of a Lie algebra $\mathfrak{g}$ by providing a complete classification of quantum subgroups of the algebras $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$. Despite a large effort since then, little progress has been made for the other simple Lie algebras. However, recent results of Gannon and Schopieray have blown this problem wide open, by providing effective bounds on the levels at which quantum subgroups can appear for a given Lie algebra $\mathfrak{g}$. Hence there has been a recent revival in the program to classify all quantum subgroups, and in particular, to construct examples that are predicted to exist. In this talk I will describe the connection between symmetries of the category of level $k$ integrable representations of $\mathfrak{g}$, and the quantum subgroups of $\mathfrak{g}$ appearing at level $k$. In particular I will give constructions of the conjectured charge conjugation quantum subgroups of $\mathfrak{sl}_n$ at all levels, and of a sporadic quantum subgroup of $\mathfrak{g}_2$ which appears at level 4.