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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Mitja Mastnak and Alexandru Nica
Journal: Trans. Amer. Math. Soc. 362 (2010), 3705-3743
MSC (2010): Primary 46L54; Secondary 16T30
Published electronically: February 8, 2010
MathSciNet review: 2601606
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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

$\displaystyle LS_{\mu \boxtimes \nu} =LS_{\mu} +LS_{\nu}, \ \forall \mu , \nu \in \mathcal{G}_k$    such that $\displaystyle \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 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

$\displaystyle 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 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.


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Additional Information

Mitja Mastnak
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Address at time of publication: Department of Mathematics and Computer Science, Saint Mary’s University, Halifax, Nova Scotia, Canada B3H 3C3
Email: mmastnak@cs.smu.ca

Alexandru Nica
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: anica@math.uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04995-0
PII: S 0002-9947(10)04995-0
Received by editor(s): August 12, 2008
Published electronically: February 8, 2010
Additional Notes: The research of the second-named author was supported by a Discovery Grant from NSERC, Canada
Article copyright: © Copyright 2010 American Mathematical Society