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The primary aim of this book is to provide an in-depth study of the Drinfeld-Jimbo quantization Uq(g) of the enveloping algebra U(g) of a semisimple Lie algebra g and of the Hopf dual Rq (G) of Uq(g). The focus is on determining the primitive spectra of these rings. A systematic use of Hopf algebra structure, and in particular of adjoint action, is made. "Quantum phenomena" - which are new features of Uq(g) - are also described. The reader will learn how the quantum viewpoint has revitalized the study of enveloping algebras and will become acquainted with proofs which have been developed over the last 20 years into a particularly efficient form.