Anyonic Symmetries and Non-Abelian Topological Defects of Bosonic Abelian Fractional Quantum (Spin) Hall States in the $ADE$ Classification*

We consider bosonic abelian Fractional Quantum Hall (FQH) and Fractional Quantum Spin Hall (FQSH) states with edge theories drawn from the $ADE$ Kac Moody algebras at level $1.$ This set of systems have `anyonic' symmetries that leave braiding and fusion invariant Remarkably, the group of anyonic symmetries for this class of models is isomorphic to the symmetries of the Dynkin diagrams of the particular ADE Lie Algebra under consideration. The triality symmetry of the Dynkin diagram of $so(8)$ leads to the largest anyonic symmetry group $S_3$ (the permutation group on 3 elements). Each element of the anyonic symmetry group corresponds to a distinct way of gapping out the edge (i.e., each element corresponds to a Lagrangian subgroup). Junctions between two distinct gapped edges host non abelian twist defects with quantum dimensions ($>1$). In the case of $so(8)$ we have more exotic twist defects with non-abelian fusion.