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Home > Vol 22, No 4 (2005)

Marcelo Aguiar and Frank Sottile

Texas A\&M University Department of Mathematics College Station TX 77843 USA College Station TX 77843 USA

Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980's by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.

1. Marcelo Aguiar, Nantel Bergeron, and Frank Sottile, “Combinatorial Hopf algebras and generalized Dehn- Sommerville relations,” math.CO/0310016. To appear in Compos. Math.
2. Marcelo Aguiar and Frank Sottile, “Structure of the Malvenuto-Reutenauer Hopf algebra of permutations,” Adv. Math. 191(2) (2005), 225-275.
3. Marcelo Aguiar and Frank Sottile, “Structure of the Hopf algebra of planar binary trees of Loday and Ronco,” math.CO/0409022,
2004. To appear in Journal of Algebra.
4. Frédéric Chapoton and Muriel Livernet, “Pre-Lie algebras and the rooted trees operad,” Internat. Math. Res. Notices (8) (2001), 395-408.
5. A. Connes and D. Kreimer, “Hopf algebras, renormalization and noncommutative geometry,” Comm. Math. Phys. 199 (1998), 203-242.
6. L. Foissy, “Les alg`ebres de Hopf des arbres enracinés décorés, I”, Bull. Sci. Math. 126(3) (2002), 193-239.
7. L. Foissy, “Les alg`ebres de Hopf des arbres enracinés décorés, II,” Bull. Sci. Math. 126(4) (2002), 249-288. AGUIAR AND SOTTILE
8. Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh, and Jean-Yves Thibon, “Noncommutative symmetric functions,” Adv. Math. 112(2) (1995), 218-348.
9. Murray Gerstenhaber and S.D. Schack, “The shuffle bialgebra and the cohomology of commutative algebras,” J. Pure Appl. Algebra 70(3) (1991), 263-272.
10. Robert Grossman and Richard G. Larson, “Hopf-algebraic structure of families of trees,” J. Algebra 126(1) (1989), 184-210.
11. Robert Grossman and Richard G. Larson, “Solving nonlinear equations from higher order derivations in linear stages,” Adv. Math. 82(2) (1990), 180-202.
12. Michiel Hazewinkel, “Generalized overlapping shuffle algebras,” Pontryagin Conference, 8, Algebra (Moscow, 1998). J. Math. Sci. (New York) 106(4) (2001), 3168-3186.
13. Michael E. Hoffman, “Quasi-shuffle products,” J. Algebraic Combin. 11(1) (2000), 49-68.
14. Michael E. Hoffman, “Combinatorics of rooted trees and Hopf algebras,” Trans. Amer. Math. Soc. 355 (2003), 3795-3811.
15. Ralf Holtkamp, “Comparison of Hopf algebras on trees,” Arch. Math. (Basel) 80(4) (2003), 368-383.
16. Samuel K. Hsiao, “Structure of the peak Hopf algebra of quasi-symmetric functions,” 2002.
17. Jean-Louis Loday, Cyclic homology, Grundlehren der Mathematischen Wissenschaften,
301. Springer-Verlag, Berlin, 1998. xx+513 pp.
18. Jean-Louis Loday and María O. Ronco, “Hopf algebra of the planar binary trees,” Adv. Math. 139(2) (1998), 293-309.
19. Jean-Louis Loday and María O. Ronco, “Order structure on the algebra of permutations and of planar binary trees,” J. Algebraic Combin. 15 (2002), 253-270.
20. Jean-Louis Loday and María O. Ronco, “On the structure of cofree Hopf algebras,” math.QA/0405330, 2004.
21. Claudia Malvenuto, “Produits et coproduits des fonctions quasi-symétriques et de l'alg`ebre des descents,” no. 16, Laboratoire de combinatoire et d'informatique mathématique (LACIM), Univ. du Québec `a Montréal, Montréal, 1994.
22. Claudia Malvenuto and Christophe Reutenauer, “Duality between quasi-symmetric functions and the Solomon descent algebra,” J. Algebra 177(3) (1995), 967-982.
23. John W. Milnor and John C. Moore, “On the structure of Hopf algebras,” Ann. of Math. 81(2) (1965), 211-264.
24. Susan Montgomery, “Hopf algebras and their actions on rings,” CBMS Regional Conference Series in Mathematics 82, American Mathematical Society, Providence, RI, 1993.
25. Frédéric Patras, “L'alg`ebre des descentes d'une big`ebre graduée,” J. Algebra 170(2) (1994), 547-566.
26. Daniel Quillen, “Rational homotopy theory,” Ann. of Math. 90(2) (1969), 205-295.
27. Christophe Reutenauer, Free lie algebras, The Clarendon Press Oxford University Press, New York, 1993, Oxford Science Publications.
28. William R. Schmitt, “Incidence Hopf algebras,” J. Pure Appl. Algebra 96(3) (1994), 299-330.
29. Manfred Schocker, “The peak algebra of the symmetric group revisited,” 2002. math.RA/0209376
30. Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original.
31. Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge University Press, Cambridge, 1999, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.
32. John R. Stembridge, “Enriched P-partitions,” Trans. Amer. Math. Soc. 349(2) (1997), 763-788.
33. Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series W. A. Benjamin, Inc., New York, 1969 vii+336 pp.