Abstract
This document is an extended abstract of the paper `Hopf algebras and the logarithm of the S-transform in free probability' in which we introduce a Hopf algebraic approach to the study of the operation $\boxtimes$ (free multiplicative convolution) from free probability.
Highlights
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 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
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.)
Summary
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. In this case the group (G1, ) can be identified as the group of characters of Sym, in such a way that the S-transform, its reciprocal 1/S and its logarithm log S relate in a natural sense to the sequences of complete, elementary and power sum symmetric functions. In [4] we connect several areas in mathematics: free probability, combinatorics of non-crossing partitions, and Hopf algebras. In this extended abstract emphasis is placed on reviewing concepts in these areas needed for understanding of our paper.
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