Hopf algebras of planar binary trees: an operated algebra approach. (English) Zbl 1442.16033
A new viewpoint is given on the Loday-Ronco Hopf algebra of planar binary trees \(H_{LR}\): in a parallel with the noncommutative Connes-Kreimer Hopf algebra of planar rooted trees which can be seen as a free operated algebra, it is seen as a free \(\vee\)-algebra: a \(\vee\)-algebra is a pair \((A,\vee)\) where \(A\) is an algebra and \(\vee\) is a bilinear (non necessarily associative) product on \(A\) such that for any \(x,x',y,y'\in A\),
\[
(x\vee x')\cdot (y\vee y')=x\vee (x'\cdot (y\vee y'))+((x\vee x')\cdot y)\vee y'.
\]
Adding a cocycle condition, one recovers the Loday-Ronco coproduct, together with a universal property for the Hopf algebra \(H_{LR}\) in the category of cocycle \(\vee\)-bialgebras. Similar results are also obtained for decorated planar binary trees.
This paper also contains a description of the Loday-Ronco coproduct in terms of admissible cuts, comparable to the combinatorial description of the Connes-Kreimer coproduct on (planar) rooted trees.
This paper also contains a description of the Loday-Ronco coproduct in terms of admissible cuts, comparable to the combinatorial description of the Connes-Kreimer coproduct on (planar) rooted trees.
Reviewer: Loïc Foissy (Calais)
MSC:
| 16T05 | Hopf algebras and their applications |
| 08B20 | Free algebras |
| 16T10 | Bialgebras |
| 16T30 | Connections of Hopf algebras with combinatorics |
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