2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59912We classify graded Hopf algebras structures over path coalgebras, that is over free pointed coalgebras, using Hopf quivers which are analogous to Cayley graphs. The description involves formulas for the product besides the canonical formulas for the coproduct. This makes explicit the quantum shuffle product making use of natural elements in a Hopf bimodule having a simple "geometrical" interpretation in the quiver sense, rather than working systematically with right or left coinvariants.12 pagesQuantum AlgebraRings and Algebras57T05;16W30text