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 Schneiders 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.
Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words.
To classify the finite dimensional pointed Hopf algebras with $G= {rm McL}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {rm GAP}.
To classify the finite dimensional pointed Hopf algebras with $G= {rm HS}$ or ${rm Co3}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {rm GAP}.
