Cumulants, free probability and Hopf algebras

Free Probability was introduced by Voiculescu 40 years ago in order to solve problems in von Neumann algebras. It has since developed into a whole new field of mathematics with intimate connections to established fields like classical probability, combinatorics and analysis, in particular random matrices , noncrossing partitions and operator algebras.

The relevance of higher algebraic structures like dual groups for free probability was observed from the very beginning of the theory and, later on, differential coalgebra structures arose in connection with the subordination phenomenon. This first period culminated in the ESI Program "Bialgebras and Free Probability", which brought together researchers from different fields in at the Erwin Schrödinger Institute in Vienna, February--April 2011.

On the other hand, the theory of free cumulants introduced by Speicher initially did not involve Hopf algebras and neither did the other notions of independence, most noteworthy Boolean and monotone, and the general framework of exchangeability systems . This has changed in the last decade and Hopf algebras came up in several guises.