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

Abstract

Let k be a positive integer and let G_k denote the set of non-commutative k-variable distributions \mu such that \mu (X_1) = ... = \mu (X_k) = 1. G_k is a group under the operation of free multiplicative convolution. We identify G_k as the group of characters of a certain Hopf algebra Y_k. Then, by using the log map from characters to infinitesimal characters of Y_k, we introduce a transform LS_{\mu} for distributions \mu in G_k. The main property of the LS-transform is that it linearizes commuting products in G_k. For \mu in G_k, the transform LS_{\mu} is a power series in k non-commuting indeterminates; 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 non-crossing partitions. In the particular case k=1 one has that Y_1 is naturally isomorphic to the Hopf algebra 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(z) = - z log S(z). In this case the group G_1 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 respectively power sum symmetric functions.