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

Let k be a positive integer and let G k denote the set of all joint distributions of k-tuples (a 1 ,..., a k ) in a noncommutative probability space (A, ϕ) such that ϕ(a 1 ) = ··· = ϕ(a k ) = 1. G k is a group under the operation of the 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 μ for distributions μ ∈ G k . LS μ is a power series in k noncommuting indeterminates z 1 ,..., z k ; its coefficients can be computed from the coefficients of the R-transform of μ by using summations over chains in the lattices NC(n) of noncrossing partitions. The LS-transform has the "linearizing" property that LS μν = LS μ + LS ν , ∀ μ , ν ∈ G k such that μ ν = ν μ. 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 μ (z) = ―z log S μ (z), ∀μ ∈ G 1 . 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.