paper

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

arXiv:0807.4169 · doi:10.1103/PhysRevLett.101.216802

Abstract

Let k be a positive integer and let G_k denote the set of non-commutative k-variable distributions μsuch that μ(X_1) = ... = μ(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_μ for distributions μin G_k. The main property of the LS-transform is that it linearizes commuting products in G_k. For μin G_k, the transform LS_μ is a power series in k non-commuting indeterminates; its coefficients can be computed from the coefficients of the R-transform of μby 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.

Version with minor revisions, to appear in Transactions of the American Mathematical Society. 39 pages, no figures

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