On the Hopf Algebra of Rooted Trees

TL;DR

This paper explores the structure of the Hopf algebra of rooted trees, providing a formula for generators, analyzing conjectures, and examining their algebraic properties and representations.

Contribution

It introduces a formula for counting generators, disproves an analogue of a known conjecture, and studies the algebraic structures and representations of these Hopf algebras.

Findings

Derived a formula for the number of generators of the filtered Hopf algebra of rooted trees.

Disproved the analogue of Andruskiewitsch and Schneider's Conjecture for these Hopf algebras.

Showed that the Hopf algebra of rooted trees and its enveloping algebra lack nonzero integrals and are local quasitriangular.

Abstract

We find a formula to compute the number of the generators, which generate the $n$-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight $n$. Applying Hopf algebra of rooted trees, we show that the analogue of Andruskiewitsch and Schneider's Conjecture is not true. The Hopf algebra of rooted trees and the enveloping algebra of the Lie algebra of rooted trees are two important examples of Hopf algebras. We give their representation and show that they have not any nonzero integrals. We structure their graded Drinfeld doubles and show that they are local quasitriangular Hopf algebras.