On a factorization of graded Hopf algebras using Lyndon words. (Q2382988)

Let \(H\) be a Hopf algebra generated by a Hopf subalgebra \(H_0\) and a vector space \(V\) such that \(\Delta V\subseteq V\otimes H_0+H_0\otimes V+H_0\otimes H_0\) and \(S(V)\subseteq KVK\). Consider the Hopf algebra filtration defined by \(\mathcal F_0=H_0\), \(\mathcal F_1=H_0+H_0VH_0\), and \(\mathcal F_n=(\mathcal F_1)^n\), and let \(\text{gr}(H)\) denote the associated graded Hopf algebra. The main results of this paper are a factorization result about \(\text{gr}(H)\), and the fact that to each factor one can associate in a natural way a graded Hopf algebra which projects onto a Nichols algebra.