This ERGEBNISSE volume provides a comprehensive and detailed account of the research in the field of quantum groups. The book is a sequel to J. Dixmier: "Algebres enveloppantes", Gauthier-Villars 1974 and J.-C. Jantzen: "Einhüllende Algebren halbeinfacher Lie-Algebren", Springer-Verlag 1983. It is an important reference work for graduate students and researchers in algebra, the theory of Lie algebras, mathematical physics.

Inhalt
I. Hopf Algebras.- 1.1 Axioms of a Hopf Algebra.- 1.2 Group Algebras and Enveloping Algebras.- 1.3 Adjoint Action.- 1.4 The Hopf Dual.- 1.5 Comments and Complements.- 2. Excerpts from the Classical Theory.- 2.1 Lie Algebras.- 2.2 Algebraic Lie Algebras.- 2.3 Algebraic Groups.- 2.4 Lie Algebras of Algebraic Groups.- 2.5 Comments and Complements.- 3. Encoding the Cartan Matrix.- 3.1 Quantum Weyl Algebras.- 3.2 The Drinfeld Double.- 3.3 The Rosso Form and the Casimir Invariant.- 3.4 The Classical Limit and the Shapovalev Form.- 3.5 Comments and Complements.- 4. Highest Weight Modules.- 4.1 The Jantzen Filtration and Sum Formula.- 4.2 Kac-Moody Lie Algebras.- 4.3 Integrable Modules for Uq(gc).- 4.4 Demazure Modules and Product Formulae.- 4.5 Comments and Complements.- 5. The Crystal Basis.- 5.1 Operators in the Crystal Limit.- 5.2 Crystals.- 5.3 Ad-invariant Filtrations, Twisted Actions and the Crystal Basis for Uq(n-).- 5.4 The Grand Loop.- 5.5 Comments and Complements.- 6. The Global Bases.- 6.1 The ? Operation and the Embedding Theorem.- 6.2 Globalization.- 6.3 The Demazure Property.- 6.4 Littelmann's Path Crystals.- 6.5 Comments and Complements.- 7. Structure Theorems for Uq(g).- 7.1 Local Finiteness for the Adjoint Action.- 7.2 Positivity of the Rosso Form.- 7.3 The Separation Theorem.- 7.4 Noetherianity.- 7.5 Comments and Complements.- 8. The Primitive Spectrum of Uq(g).- 8.1 The Poincaré Series of the Harmonic Space.- 8.2 Factorization of the Quantum PRV Determinants.- 8.3 Verma Module Annihilators.- 8.4 Equivalence of Categories.- 8.5 Comments and Complements.- 9. Structure Theorems for Rq[G].- 9.1 Commutativity Relations.- 9.2 Surjectivity and Injectivity Theorems.- 9.3 The Adjoint Action.- 9.4 The R-Matrix.- 9.5 Comments and Complements.- 10. The PrimeSpectrum of Rq[G].- 10.1 Highest Weight Modules.- 10.2 The Quantum Weyl Group.- 10.3 Prime and Primitive Ideals of Rq[G].- 10.4 Hopf Algebra Automorphisms.- 10.5 Comments and Complements.- A.2 Excerpts from Ring Theory.- A.3 Combinatorial Identities and Dimension Theory.- A.4 Remarks on Constructions of Quantum Groups.- A.5 Comments and Complements.- Index of Notation.