Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. M. Aguiar, N. Bergeron, F. Sottile. Combinatorial Hopf algebras and generalized Dehn-Somerville equations, Compositio Mathematica 142 (2006), 1-30. MR 2196760 (2006h:05233)
  • 2. M. Anshelevich, E.G. Effros, M. Popa. Zimmermann type cancellation in the free Faà di Bruno algebra, Journal of Functional Analysis 237 (2006), 76-104. MR 2239259
  • 3. K. Dykema. Multilinear function series and transforms in free probability theory, Advances in Mathematics 208 (2007), 351-407. MR 2304321 (2008k:46193)
  • 4. R. Ehrenborg. On posets and Hopf algebras, Advances in Mathematics 119 (1996), 1-25. MR 1383883 (97e:16079)
  • 5. H. Figueroa, J.M. Gracia-Bondia. Combinatorial Hopf algebras in quantum field theory I, Reviews in Mathematical Physics 17 (2005), 881-976. MR 2167639 (2006g:16085)
  • 6. U. Haagerup. On Voiculescu's $ R$- and $ S$-transforms for free non-commuting random variables, in Free Probability Theory (D. Voiculescu, editor), Fields Institute Communications 12 (1997), 127-148. MR 1426838 (98c:46137)
  • 7. S.A. Joni, G.-C. Rota. Coalgebras and bialgebras in combinatorics, Studies in Applied Mathematics 61 (1979), 93-139. MR 544721 (81c:05002)
  • 8. G. Kreweras. Sur les partitions non-croisées d'un cycle, Discrete Mathematics 1 (1972), 333-350. MR 0309747 (46:8852)
  • 9. I.G. MacDonald. Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979. MR 553598 (84g:05003)
  • 10. A. Nica, R. Speicher. A ``Fourier transform'' for multiplicative functions on non-crossing partitions, Journal of Algebraic Combinatorics 6 (1997), 141-160. MR 1436532 (98i:46070)
  • 11. A. Nica, R. Speicher. Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series 335, Cambridge University Press, 2006. MR 2266879 (2008k:46198)
  • 12. W.R. Schmitt. Incidence Hopf algebras, Journal of Pure and Applied Algebra 96 (1994), 299-330. MR 1303288 (95m:16033)
  • 13. R. Simion. Noncrossing partitions, Discrete Mathematics 217 (2000), 367-409. MR 1766277 (2001g:05011)
  • 14. R.P. Stanley. Enumerative combinatorics volume 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999. MR 1676282 (2000k:05026)
  • 15. M.E. Sweedler. Hopf algebras, Benjamin, 1969. MR 0252485 (40:5705)
  • 16. D. Voiculescu. Multiplication of certain noncommuting random variables, Journal of Operator Theory 18 (1987), 223-235. MR 915507 (89b:46076)
  • 17. D.V. Voiculescu, K.J. Dykema, A. Nica. Free random variables, CRM Monograph Series 1, American Mathematical Society, 1992. MR 1217253 (94c:46133)
  • 18. A.V. Zelevinsky. Representations of finite classical groups. A Hopf algebra approach, Lecture Notes in Mathematics 869, Springer Verlag, 1981. MR 643482 (83k:20017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46L54, 16T30

Retrieve articles in all journals with MSC (2010): 46L54, 16T30

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

Alexandru Nica
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

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

American Mathematical Society