Hopf algebras and the logarithm of the S-transform in free probability – Extended abstract

Two basic tools of free probability are the R-transform and the S-transform. These transforms were introduced by Voiculescu in the 1980s, and are used to understand the addition and the multiplication of two free random variables respectively. The R-transform has a natural and very useful multi-variable extension describing the addition of two free k-tuples of random variables, but the problem of finding such an extension for the S-transform is open. The problem of the multi-variable S-transform can be re-phrased as the problem of understanding the structure of the group (Gk, ), where Gk is a special set of joint distributions of noncommutative k-tuples (see precise definition in Equation (3.3) below), and where (“free multiplicative convolution”) is a binary operation on Gk which encodes the multiplication of free k-tuples. At present, the structure of (Gk, ) is well-understood only in the special case k = 1; in this case, the S-transform of Voiculescu provides an isomorphism between G1 and a multiplicative group of power series in one variable. (A word of caution here: G1 is commutative, but it is easy to see that Gk is not commutative for any k ≥ 2.) In [4] we use Hopf algebra methods in order to study the multiplication of free k-tuples. Specifically, we construct a combinatorial Hopf algebra Y such that (Gk, ) is naturally isomorphic to the the group X(Y) of characters of Y. We then employ the log map from characters to infinitesimal characters of Y, to introduce a transform LSμ for distributions μ ∈ Gk. LSμ is a power series in k non-commuting indeterminates z1, . . . , zk; its coefficients can be computed from the coefficients of the R-transform of μ †Research supported by a Discovery Grant from NSERC, Canada.