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Shuffle bialgebras. (English. French summary) Zbl 1239.16032
The author proposes a framework in which to obtain the primitive elements in various combinatorial Hopf algebras over a field $$K$$. If $$n=n_1+n_2+\cdots+n_r$$ is a composition of $$n$$, let $$Sh(n_1,n_2,\dots,n_r)$$ denote the usual shuffles of $$\{1,2,\dots,n\}$$ determined by the composition. A shuffle algebra is a vector space graded by the positive integers, with linear maps $$\cdot_g\colon A_n\otimes A_m\to A_{n+m}$$ for each $$g$$ in $$Sh(n,m)$$ satisfying associative-type conditions. The reduced tensor algebra $$T(V)/K$$ on a vector space $$V$$ is a shuffle algebra. For a positively graded vector space $$V$$, there is a free shuffle algebra $$Sh(V)$$ on $$V$$. A source of shuffle algebras is nonunital infinitesimal bialgebras, as introduced by J.-L. Loday and the author [J. Reine Angew. Math. 592, 123-155 (2006; Zbl 1096.16019)]. These are positively graded algebras with a coassociative coproduct $$c$$ satisfying $$c(xy)=(xy_1)\otimes y_2+x_1\otimes (x_2)y+x\otimes y$$ (Sweedler notation). They admit natural structures as shuffle algebras. A shuffle bialgebra is a positively graded shuffle algebra with a graded coassociative coproduct satisfying a rule for the coproduct of a product $$x \cdot_g y$$. Free shuffle algebras on graded coalgebras are shuffle bialgebras. If $$A$$ is a graded nonunital infinitesimal bialgebra, its shuffle algebra structure is a shuffle bialgebra (same coproduct). Primitive elements enter for $$H$$ a conilpotent infinitesimal bialgebra, where conilpotent means that the union of the ascending filtration of $$H$$ starting with $$\text{Prim}(H)$$ is $$H$$.
A principal theorem says that $$H$$ is isomorphic to a certain enveloping-type algebra of its primitives. This theorem is proved in a more general context of preshuffle bialgebras and pre-Lie systems. All this is intertwined with and applied to various combinatorial Hopf algebras, including maps between finite sets, the Malvenuto-Reutenauer bialgebra of permutations, the bialgebra of surjective maps (of finite sets), the bialgebra of parking functions, and planar rooted trees. The primitive elements of some of these were computed earlier, but here this study is done in a very general framework.

MSC:
 16T10 Bialgebras 16T30 Connections of Hopf algebras with combinatorics 05E15 Combinatorial aspects of groups and algebras (MSC2010)
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References:
 [1] Aguiar, Marcelo, Hopf algebras, 237, Infinitesimal bialgebras, pre-Lie and dendriform algebras, 1-33, (2004), Dekker, New York · Zbl 1059.16027 [2] Aguiar, Marcelo; Sottile, Frank, Structure of the malvenuto-reutenauer Hopf algebra of permutations, Adv. Math., 191, 2, 225-275, (2005) · Zbl 1056.05139 [3] Aguiar, Marcelo; Sottile, Frank, Structure of the loday-ronco Hopf algebra of trees, J. Algebra, 295, 2, 473-511, (2006) · Zbl 1099.16015 [4] Bergeron, F.; Bergeron, N.; Howlett, R. B.; Taylor, D. E., A decomposition of the descent algebra of a finite Coxeter group, J. Algebraic Combin., 1, 1, 23-44, (1992) · Zbl 0798.20031 [5] Duchamp, Gérard; Hivert, Florent; Thibon, Jean-Yves, Noncommutative symmetric functions. VI. free quasi-symmetric functions and related algebras, Internat. J. Algebra Comput., 12, 5, 671-717, (2002) · Zbl 1027.05107 [6] Gerstenhaber, Murray, The cohomology structure of an associative ring, Ann. of Math. (2), 78, 267-288, (1963) · Zbl 0131.27302 [7] Ginzburg, Victor; Kapranov, Mikhail, Koszul duality for operads, Duke Math. J., 76, 1, 203-272, (1994) · Zbl 0855.18006 [8] Hivert, Florent; Novelli, Jean-Christophe; Thibon, Jean-Yves, Un analogue du monoïde plaxique pour LES arbres binaires de recherche, C. R. Math. Acad. Sci. Paris, 335, 7, 577-580, (2002) · Zbl 1013.05026 [9] Holtkamp, Ralf, On Hopf algebra structures over free operads, Adv. Math., 207, 2, 544-565, (2006) · Zbl 1117.16027 [10] Livernet, Muriel, From left modules to algebras over an operad: application to combinatorial Hopf algebras, Ann. Math. Blaise Pascal, x+49 pp., (2009) · Zbl 1206.18010 [11] Loday, Jean-Louis, Dialgebras and related operads, 1763, Dialgebras, 7-66, (2001), Springer, Berlin · Zbl 0999.17002 [12] Loday, Jean-Louis, Generalized bialgebras and triples of operads, Astérisque, 320, x+116 pp., (2008) · Zbl 1178.18001 [13] Loday, Jean-Louis; Ronco, María, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $$K$$-theory, 346, Trialgebras and families of polytopes, 369-398, (2004), Amer. Math. Soc., Providence, RI · Zbl 1065.18007 [14] Loday, Jean-Louis; Ronco, María, On the structure of cofree Hopf algebras, J. Reine Angew. Math., 592, 123-155, (2006) · Zbl 1096.16019 [15] Markl, Martin; Shnider, Steve; Stasheff, Jim, Operads in algebra, topology and physics, 96, (2002), American Mathematical Society, Providence, RI · Zbl 1017.18001 [16] Novelli, Jean-Christophe; Thibon, Jean-Yves, Hopf algebras and dendriform structures arising from parking functions, Fund. Math., 193, 3, 189-241, (2007) · Zbl 1127.16033 [17] Novelli, Jean-Christophe; Thibon, Jean-Yves, Parking functions and descent algebras, Ann. Comb., 11, 1, 59-68, (2007) · Zbl 1115.05095 [18] Palacios, Patricia; Ronco, María O., Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra, 299, 2, 648-678, (2006) · Zbl 1110.16046 [19] Patras, Frédéric; Schocker, Manfred, Twisted descent algebras and the Solomon-Tits algebra, Adv. Math., 199, 1, 151-184, (2006) · Zbl 1154.16029 [20] Pirashvili, Teimuraz, Sets with two associative operations, Cent. Eur. J. Math., 1, 2, 169-183 (electronic), (2003) · Zbl 1032.16032 [21] Ronco, María, Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. Algebra, 254, 1, 152-172, (2002) · Zbl 1017.16033 [22] Solomon, Louis, A MacKey formula in the group ring of a Coxeter group, J. Algebra, 41, 2, 255-264, (1976) · Zbl 0355.20007
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