# Braided dendriform and tridendriform algebras and braided Hopf algebras of planar trees

## TL;DR

This paper develops braided algebraic structures on planar trees, extending known Hopf algebras with braidings, and explores their properties and isomorphisms in the braided context.

## Contribution

It introduces braided dendriform and tridendriform algebras, extending Hopf algebras of planar trees with braidings and establishing new isomorphisms.

## Key findings

- Braided dendriform and tridendriform algebras constructed.
- Extended Hopf algebra isomorphisms to braided settings.
- New braiding variations for planar rooted forests.

## Abstract

This paper introduces the braidings of dendriform algebras and tridendriform algebras. By studying free braided dendriform algebras, we obtain braidings of the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also give a variation of the braiding of Foissy for the noncommutative Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted forests so that the well-known isomorphism between this Hopf algebra and the Loday-Ronco Hopf algebra is extended to the braided context. As free braided tridendriform algebras, we also give braided extension of the Hopf algebra of Loday and Ronco on planar rooted trees.