Bidendriform bialgebras, trees, and free quasi-symmetric functions. (English) Zbl 1123.16030
A dendriform algebra \(A\) is a (non-unitary) algebra whose associative product splits into two parts \(\prec\) and \(\succ\) such that \((A,\prec,\succ)\) is an \(A\)-bimodule [cf. J.-L. Loday, Lect. Notes Math. 1763, 7-66 (2001; Zbl 0999.17002)]. There is a dual notion of (non-counital) dendriform coalgebra. A dendriform bialgebra is a dendriform algebra which is a coalgebra satisfying compatibility conditions [cf. J.-L. Loday and M. O. Ronco, Adv. Math. 139, No. 2, 293-309 (1998; Zbl 0926.16032)].
In the paper under review, the author introduces a bidendriform bialgebra as a dendriform bialgebra whose comultiplication gives it the structure of a dendriform coalgebra and satisfying compatibility conditions. The paper compares FQSym, the Hopf algebra of free quasi-symmetric functions, with \(H^D\), the Hopf algebra of planar rooted trees with decoration \(D\). FQSym is the Malvenuto-Reutenauer Hopf algebra [J.-L. Loday and M. O. Ronco, loc. cit.]. \(H^D\) was introduced by the author [Bull. Sci. Math. 126, No. 3, 193-239 (2002; Zbl 1013.16026) and ibid. No. 4, 249-288 (2002; Zbl 1013.16027)]. In Section 4 of the paper under review, FQSym (or more precisely, its augmentation ideal) is given the structure of a bidendriform bialgebra.
Putting a bidendriform bialgebra structure on \(H^D\) (or again its augmentation ideal) is more complicated. It involves the notion of a dendriform module, and an altered version of tensor product for dendriform algebras. The ultimate result is that there is a graded (by the natural numbers \(\mathbb{N}\)) set \(D\) such that the augmentation ideals of FQSym and \(H^D\) are isomorphic as \(\mathbb{N}\)-graded Hopf algebras. The author had proved earlier [loc. cit.] that the primitive elements (at characteristic zero) of \(H^D\) form a free Lie algebra. Thus the same result holds for FQSym, proving a conjecture of G. Duchamp, F. Hivert and J.-Y. Thibon [in Formal power series and algebraic combinatorics. Proc. 12th int. Conf., FPSAC’00, Moscow, Russia. Berlin: Springer. 170-178 (2000; Zbl 0958.05131)].
In the paper under review, the author introduces a bidendriform bialgebra as a dendriform bialgebra whose comultiplication gives it the structure of a dendriform coalgebra and satisfying compatibility conditions. The paper compares FQSym, the Hopf algebra of free quasi-symmetric functions, with \(H^D\), the Hopf algebra of planar rooted trees with decoration \(D\). FQSym is the Malvenuto-Reutenauer Hopf algebra [J.-L. Loday and M. O. Ronco, loc. cit.]. \(H^D\) was introduced by the author [Bull. Sci. Math. 126, No. 3, 193-239 (2002; Zbl 1013.16026) and ibid. No. 4, 249-288 (2002; Zbl 1013.16027)]. In Section 4 of the paper under review, FQSym (or more precisely, its augmentation ideal) is given the structure of a bidendriform bialgebra.
Putting a bidendriform bialgebra structure on \(H^D\) (or again its augmentation ideal) is more complicated. It involves the notion of a dendriform module, and an altered version of tensor product for dendriform algebras. The ultimate result is that there is a graded (by the natural numbers \(\mathbb{N}\)) set \(D\) such that the augmentation ideals of FQSym and \(H^D\) are isomorphic as \(\mathbb{N}\)-graded Hopf algebras. The author had proved earlier [loc. cit.] that the primitive elements (at characteristic zero) of \(H^D\) form a free Lie algebra. Thus the same result holds for FQSym, proving a conjecture of G. Duchamp, F. Hivert and J.-Y. Thibon [in Formal power series and algebraic combinatorics. Proc. 12th int. Conf., FPSAC’00, Moscow, Russia. Berlin: Springer. 170-178 (2000; Zbl 0958.05131)].
Reviewer: Earl J. Taft (New Brunswick)
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 17A30 | Nonassociative algebras satisfying other identities |
| 05C05 | Trees |
| 05E05 | Symmetric functions and generalizations |
Keywords:
bidendriform bialgebras; planar rooted trees; free quasi-symmetric functions; dendriform coalgebras; Malvenuto-Reutenauer Hopf algebras; augmentation ideals; graded Hopf algebrasOnline Encyclopedia of Integer Sequences:
Number of free generators of degree n of the primitive Lie algebra of the Hopf algebra of Free quasi-symmetric functions (or Malvenuto-Reutenauer algebra of permutations).Number of independent generators of degree n of the algebra of Free quasi-symmetric functions (or Malvenuto-Reutenauer algebra of permutations) as a dendriform dialgebra (i.e., number of totally primitive elements).
References:
| [1] | Aguiar, Marcelo, (Infinitesimal Bialgebras, Pre-Lie and Dendriform Algebras. Infinitesimal Bialgebras, Pre-Lie and Dendriform Algebras, Lecture Notes in Pure and Appl. Math., vol. 237 (2004), Dekker: Dekker New York) · Zbl 1059.16027 |
| [2] | Aguiar, Marcelo; Orellana, Rosa C., The Hopf algebra of uniform block permutations (2005) · Zbl 1180.16024 |
| [3] | Aguiar, Marcelo; Sottile, Frank, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Adv. Math., 191, 2, 225-275 (2005) · Zbl 1056.05139 |
| [4] | Connes, Alain; Kreimer, Dirk, Hopf algebras, Renormalization and Noncommutative geometry, Comm. Math. Phys., 199, 1, 203-242 (1998), hep-th/98 08042 · Zbl 0932.16038 |
| [5] | Duchamp, Gérard; Hivert, Florent; Thibon, Jean-Yves, Some Generalizations of Quasi-symmetric Functions and Noncommutative Symmetric Functions (2000), Springer: Springer Berlin · Zbl 0958.05131 |
| [6] | Loïc Foissy, Les algèbres de Hopf des arbres enracinés décorés, Thèse de doctorat, Université de Reims, 2002; Loïc Foissy, Les algèbres de Hopf des arbres enracinés décorés, Thèse de doctorat, Université de Reims, 2002 |
| [7] | Foissy, Loïc, Les algèbres de Hopf des arbres enracinés, I, Bull. Sci. Math., 126, 193-239 (2002) · Zbl 1013.16026 |
| [8] | Foissy, Loïc, Les algèbres de Hopf des arbres enracinés, II, Bull. Sci. Math., 126, 249-288 (2002) · Zbl 1013.16027 |
| [9] | Holtkamp, Ralf, Comparison of Hopf algebras on trees, Arch. Math. (Basel), 80, 4, 368-383 (2003) · Zbl 1056.16030 |
| [10] | Kreimer, Dirk, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys., 2, 2, 303-334 (1998), q-alg/97 07029 · Zbl 1041.81087 |
| [11] | Kreimer, Dirk, On overlapping divergences, Comm. Math. Phys., 204, 3, 669-689 (1999), hep-th/98 10022 · Zbl 0977.81091 |
| [12] | Kreimer, Dirk, Combinatorics of (perturbative) quantum field theory, Phys. Rep., 4-6, 387-424 (2002), hep-th/00 10059 · Zbl 0994.81080 |
| [13] | Livernet, Muriel, A rigidity theorem for prelie algebras (2005) · Zbl 1134.17001 |
| [14] | Jean-Louis Loday, Generalized bialgebras and triples of operads, available at http://www-irma.u-strasbg.fr/ loday/; Jean-Louis Loday, Generalized bialgebras and triples of operads, available at http://www-irma.u-strasbg.fr/ loday/ · Zbl 1178.18001 |
| [15] | Loday, Jean-Louis, (Dialgebras. Dialgebras, Lecture Notes in Math., vol. 1763 (2001), Springer: Springer Berlin) · Zbl 0999.17002 |
| [16] | Loday, Jean-Louis, Arithmetree, J. Algebra, 258, 1, 275-309 (2002) · Zbl 1063.16044 |
| [17] | Jean-Louis Loday, Scindement d’associativité et algèbres de Hopf, Actes des Journées Mathématiques in la Mémoire de Jean Leray, Sémin. Congr., vol. 9, Soc. Math. France, Paris, 2004, pp. 155-172; Jean-Louis Loday, Scindement d’associativité et algèbres de Hopf, Actes des Journées Mathématiques in la Mémoire de Jean Leray, Sémin. Congr., vol. 9, Soc. Math. France, Paris, 2004, pp. 155-172 |
| [18] | Loday, Jean-Louis; Ronco, Maria O., Hopf algebra of the planar binary trees, Adv. Math., 139, 2, 293-309 (1998) · Zbl 0926.16032 |
| [19] | Loday, Jean-Louis; Ronco, Maria O., Order structure on the algebra of permutations and of planar binary trees, J. Algebraic Combin., 15, 3, 253-270 (2002) · Zbl 0998.05013 |
| [20] | Malvenuto, Claudia; Reutenauer, Christophe, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177, 3, 967-982 (1995) · Zbl 0838.05100 |
| [21] | Markl, Martin; Shnider, Steve; Stasheff, Jim, (Operads in Algebra, Topology and Physics. Operads in Algebra, Topology and Physics, Mathematical Surveys and Monographs, vol. 96 (2002), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1017.18001 |
| [22] | Milnor, John W.; Moore, John C., On the structure of Hopf algebras, Ann. of Math., 81, 2, 211-264 (1965) · Zbl 0163.28202 |
| [23] | Novelli, Jean-Christophe; Thibon, Jean-Yves, A Hopf algebra of parking functions (2003) · Zbl 1029.05033 |
| [24] | Novelli, Jean-Christophe, Hopf algebras and dendriform structures arising from parking functions (2005) · Zbl 1127.16033 |
| [25] | Ronco, Maria O., Primitive elements of a free dendriform algebra, Contemp. Math., 267, 245-263 (2000) · Zbl 0974.16035 |
| [26] | Ronco, Maria O., Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. Algebra, 254, 1, 152-172 (2002) · Zbl 1017.16033 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
