Seminario de Análisis Funcional "Mischa Cotlar" Classification of subfactors of small index

Seminario de Análisis Funcional “Mischa Cotlar” Classification of subfactors of small index

Expositor: Vaughan Jones (Vanderbilt University, Estados Unidos)

viernes 28 de febrero de 2014 a las 14 hs.
Auditorio Alberto González Domínguez
Saavedra 15 3º piso


A subfactor is an inclusion $N \subset M$ of von Neumann algebras with trivial centers.  The simplest example comes from the fixed points of a group action $M^G \subset M$, and subfactors can be thought of as fixed points of more general group-like algebraic structures.  These algebraic structures are closely related to tensor categories and have played important roles in knot theory, quantum groups, statistical mechanics, and topological quantum field theory.  There’s a measure of size of a subfactor, called the index.  Remarkably the values of the index below 4 are quantized, which suggests that it may be possible to classify subfactors of small index.  Subfactors of index at most 4 were classified in the ’80s and early ’90s.  The possible index values above 4 are not quantized, but once you exclude a certain family, it turns out that again the possibilities are quantized.  Recently, the classification of subfactors has been extended up to index 5, and (outside of the infinite families) there are only 10 subfactors of index between 4 and 5.  We give a summary of the key ideas in this classification and discuss what is known about these special small subfactors.