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Hopf algebra of the planar binary trees. (English) Zbl 0926.16032
Let \(k[S_n]\) be the group algebra of the symmetric group \(S_n\) over the field \(k\). The Solomon descent algebra \(Sol_n\) is a subalgebra of dimension \(2^{n-1}\) of \(k[S_n]\), and has a basis \(Q_n\) such that the linear dual of the composite map \(k[Q_n]\simeq Sol_n\to k[S_n]\) is induced by a set-theoretic map \(S_n\to Q_n\). It is known that a graded Hopf algebra structure on \(k[S_\infty]=\bigoplus_{n\geq 0}k[S_n]\) can be constructed such that \(Sol_\infty=\bigoplus_{n\geq 0}Sol_n\) is a Hopf subalgebra. There is a set \(Y_n\), made of planar binary trees with \(n\) vertices, such that the projection from \(S_n\) to \(Q_n\) is the composite of two maps \(\phi\colon Y_n\to Q_n\) and \(\psi\colon S_n\to Y_n\). The main result of the paper is to show that the graded vector space \(k[Y_\infty]=\bigoplus_{n\geq 0}k[Y_n]\) may be equipped with a structure of a graded Hopf algebra such that \(\phi\) and \(\psi\) induce Hopf algebra morphisms.

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
20C30 Representations of finite symmetric groups
16W50 Graded rings and modules (associative rings and algebras)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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