The invention of quantum groups is one of the outstanding achievements of mathematical physics and mathematics in the late twentieth century. The birth of the new theory and its rapid development are results of a strong interrelation between mathematics and physics. Quantu~ groups arose in the work of L.D. Faddeev and the Leningrad school on the inverse scattering method in order to solve integrable models. The algebra Uq(sh) appeared first in 1981 in a paper by P.P. Kulish and N.Yu. Reshetikhin on the study of integrable XYZ models with highest spin. Its Hopf algebra structure was discovered later by E.K. Sklyanin. A major event was the discovery by V.G. Drinfeld and M. Jimbo around 1985 of a class of Hopf algebras which can be considered as one-parameter deforma­ tions of universal enveloping algebras of semisimple complex Lie algebras. These Hopf algebras will be called Drinfeld-Jimbo algebras in this book. Al­ most simultaneously, S.L. Woronowicz invented the quantum group SUq(2) and developed his theory of compact quantum matrix groups. An algebraic approach to quantized coordinate algebras was given about this time by Yu.I. Manin.

I. An Introduction to Quantum Groups.- 1. Hopf Algebras.- 2. q-Calculus.- 3. The Quantum Algebra Uq(sl2) and Its Representations.- 4. The Quantum Group SLq(2) and Its Representations.- 5. The q-Oscillator Algebras and Their Representations.- II. Quantized Universal Enveloping Algebras.- 6. Drinfeld-Jimbo Algebras.- 7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras.- 8. Quasitriangularity and Universal R-Matrices.- III. Quantized Algebras of Functions.- 9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces.- 10. Coquasitriangularity and Crossed Product Constructions.- 11. Corepresentation Theory and Compact Quantum Groups.- IV. Noncommutative Differential Calculus.- 12. Covariant Differential Calculus on Quantum Spaces.- 13. Hopf Bimodules and Exterior Algebras.- 14. Covariant Differential Calculus on Quantum Groups.