2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210337By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known construction by (generalized) quantum doubles, but unlike in the known situation, it saves us from difficulties in checking complicated defining relations.To appear in a theme issue of the Arabian Journal for Science and Engineering. The published version will renumber the sections so that for example, Theorem 3.10 in this version will be called Theorem 4.10Quantum AlgebraRings and Algebras16W30, 17B37text