2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143818Andruskiewitsch and Schneider classify a large class of pointed Hopf algebras with abelian coradical. The quantum double of each such Hopf algebra is investigated. The quantum doubles of a family of Hopf algebras from the above classification are ribbon Hopf algebras.Some details are added. Corollary 2.4 and the remark after it are reformulated. Section 4 with a general tareatment is added. References are updatedRings and AlgebrasQuantum Algebra16W35, 16W40text