Subfactors and Quantum Symmetries

The theory of subfactors was initiated by V. Jones in the eighties as a Galois theory for inclusions of von Neumann algebras. Subfactors can be used to capture symmetries of various mathematical and physical objects. Part of this structure can be uncovered by computing the standard invariant of the subfactor. This is a group-like object naturally associated to the subfactor, consisting in commuting squares of inclusions of finite dimensional matrix algebras. We present some recent finiteness results for commuting squares and standard invariants. Applications to groups, Hopf algebras and Hadamard matrices are also discussed.