Finite dimensional comodules over the Hopf algebra of rooted trees (Q1858179)

Let \({\mathcal H}_{\mathcal R}\) denote the Hopf algebra of rooted trees. The author proves some results about finite-dimensional comodules over \({\mathcal H}_{\mathcal R}\). The author first shows that if \(C\) is such a comodule of dimension \(n\) then for each \(i\in\{1,2,\dots,n\}\) there exists a submodule \(C^{(i)}\) of dimension \(i\) with \(C^{(1)}\subset\cdots\subset C^{(n)}=C\). Using this he provides a parametrization of such comodules by certain finite families of primitive elements and proves a classification results for such comodules arising from certain restricted families of primitive elements. He then proves that the Lie algebra of rooted trees is free. Next, the author considers the subalgebra of \({\mathcal H}_{\mathcal R}\) generated by ladders and shows that a basis of the primitive elements can be obtained by an inductive process. In the rest of the paper the author studies endomorphisms of \({\mathcal H}_{\mathcal R}\).