  We study symmetries and defects of a wide class of two dimensional Abelian
topological phases characterized by Lie algebras. We formulate the symmetry
group of all Abelian topological field theories. The symmetries relabel
quasiparticles (or anyons) but leave exchange and braiding statistics
unchanged. Within the class of $ADE$ phases in particular, these anyonic
symmetries have a natural origin from the Lie algebra. We classify one
dimensional gapped phases along the interface between identical topological
states according to symmetries. This classification also applies to gapped
edges of a wide range of fractional quantum spin Hall(QSH) states. We show that
the edge states of the $ADE$ QSH systems can be gapped even in the presence of
time reversal and charge conservation symmetry. We distinguish topological
point defects according to anyonic symmetries and bound quasiparticles.
Although in an Abelian system, they surprisingly exhibit non-Abelian fractional
Majorana-like characteristics from their fusion behavior.
