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Structure of the Loday-Ronco Hopf algebra of trees. (English) Zbl 1099.16015
Let \(LR\) denote the graded Hopf algebra structure on the linear span of the set of rooted planar binary trees as defined and developed by J.-L. Loday and M. O. Ronco [Adv. Math. 139, No. 2, 293–309 (1998; Zbl 0926.16032); J. Algebr. Comb. 15, No. 3, 253–270 (2002; Zbl 0998.05013)]. The paper under review studies the graded Hopf algebra structure of \(YSym=(LR)^*\), the graded dual Hopf algebra of \(LR\), and relates it to various other Hopf algebras, in particular \(SSym\), the Malvenuto-Reutenauer Hopf algebra of permutations [C. Malvenuto and C. Reutenauer, J. Algebra 177, No. 3, 967–982 (1995; Zbl 0838.05100)]; \(NSym\), the Hopf algebra of noncommutative symmetric functions [I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, and J.-Y. Thibon, Adv. Math. 112, No. 2, 218–348 (1995; Zbl 0831.05063)] and \(QSym\), the Hopf algebra of quasisymmetric functions.
They construct a new basis of \(YSym\) related to the basis of \(LR\) given by Loday and Ronco via Möbius inversion on the poset of trees, and use it to study the elementary structure of \(YSym\). In particular, a geometric interpretation of the product is given, and a description of the coproduct on their basis leads to a new grading for which \(YSym\) is cofree (as graded coalgebra). This basis also leads to an explicit isomorphism between \(LR\) and the noncommutative Connes-Kreimer Hopf algebra of L. Foissy [Bull. Sci. Math. 126, No. 3, 193–239 (2002; Zbl 1013.16026) and ibid. 126, No. 4, 249–288 (2002; Zbl 1013.16027)]. This isomorphism is used to show that a canonical involution of \(QSym\) can be lifted to \(YSym\), and to construct a commutative diagram involving the Connes-Kreimer Hopf algebras (commutative and noncommutative) on one hand, and symmetric and noncommutative symmetric functions on the other hand.

MSC:
16T05 Hopf algebras and their applications
16T30 Connections of Hopf algebras with combinatorics
05E05 Symmetric functions and generalizations
16W50 Graded rings and modules (associative rings and algebras)
05C05 Trees
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