Quantum Symmetries of von Neumann algebras

Abstract: Von Neumann algebras were originally developed as a framework for representation theory and quantum mechanics. They possess rich "quantum symmetries" which generalize the symmetries witnessed by groups. These symmetries can be described by using von Neumann subalgebras or tensor categories. Jones showed that the fundamental symmetries associated to subalgebras are captured by the so-called Temperley-Lieb-Jones algebras. Soon afterwards, Bisch and Jones identified the corresponding symmetry in the presence of a single intermediate subalgebra. However, the structure of two intermediate subalgebras is much more subtle: Its structure depends strongly on their relative position, and little progress has been made over the past two decades. In this talk, we give a gentle introduction to von Neumann algebras and their quantum symmetry. Then we present the classification of the fundamental symmetry associated to two intermediate subalgebras satisfying certain group-like conditions. This talk is based on joint work with Dietmar Bisch.