Hopf algebras and the logarithm of the $S$-transform in free probability

Abstract

Let $k$ be a positive integer and let $\mathcal {G}_k$ denote the set of all joint distributions of $k$-tuples $(a_1,\ldots ,a_k)$ in a noncommutative probability space $(\mathcal {A},\varphi )$ such that $\varphi (a_1)=\cdots =\varphi (a_k) = 1$. $\mathcal {G}_k$ is a group under the operation of the free multiplicative convolution $\boxtimes$. We identify $\bigl ( \mathcal {G}_k, \boxtimes \bigr )$ as the group of characters of a certain Hopf algebra $\mathcal {Y}^{(k)}$. Then, by using the log map from characters to infinitesimal characters of $\mathcal {Y}^{(k)}$, we introduce a transform $LS_{\mu }$ for distributions $\mu \in \mathcal {G}_k$. $LS_{\mu }$ is a power series in $k$ noncommuting indeterminates $z_1, \ldots , z_k$; its coefficients can be computed from the coefficients of the $R$-transform of $\mu$ by using summations over chains in the lattices $NC(n)$ of noncrossing partitions. The $LS$-transform has the “linearizing” property that \[ LS_{\mu \boxtimes \nu } =LS_{\mu } +LS_{\nu }, \ \ \forall \mu , \nu \in \mathcal {G}_k \mbox { such that } \mu \boxtimes \nu = \nu \boxtimes \mu . \] In the particular case $k=1$ one has that ${\mathcal Y}^{(1)}$ is naturally isomorphic to the Hopf algebra $\mbox {Sym}$ of symmetric functions and that the $LS$-transform is very closely related to the logarithm of the $S$-transform of Voiculescu by the formula \[ LS_{\mu } (z) = -z \log S_{\mu } (z), \ \ \forall \mu \in \mathcal {G}_1. \] In this case the group $(\mathcal G_1, \boxtimes )$ can be identified as the group of characters of $\mbox {Sym}$, 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, respectively, power sum symmetric functions.