2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/153486The double quantum groups are the Hopf algebras underlying the complex quantum groups of which the simplest example is the quantum Lorentz group. They are non- standard quantizations of the double group $G \times G$. We construct a corresponding quantized universal enveloping algebra (QUE) and prove that the pairing between a quantum double group and its QUE is nondegenerate. We analyze the representation theory of these double quantum groups, give a detailed version of the Iwasawa decomposition proved by Podles and Woronowicz for the quantum Lorentz group, and show that they are noetherian algebras. Finally we outline a construction of more general non-standard quantum groups using quantum double groups and their generalizations.AMS-Latex, 17 pagesQuantum Algebratext