Sections 1.2 and 1.3 contain the main general definitions and notions on Hopf algebras needed in the text. The exposition is organized such that different parts of the text can be read and used (almost) independently of others. The aim of this book is to provide a treatment of the theory of quantum algebras (quantized universal enveloping algebras), quantum groups (quantized algebras of functions), their representations and corepresentations, and the noncommutative differential calculus on quantum groups. In this book at least three such classes are extensively studied: quasitriangular Hopf algebras, coquasitriangular Hopf algebras, and compact quantum group algebras. Instead of searching for a rigorous definition of a quantum group it seems to be more fruitful to look for classes of Hopf algebras that give rise to a rich theory with important applications and con. It is commonly accepted that quantum groups are certain 'nice' Hopf algebras and that the standard deformations of the enveloping Hopf algebras of semisimple Lie algebras and of coordinate Hopf algebras of the corresponding Lie groups are guiding eiamples. Currently there is no satisfactory general definition of a quantum group. It is expected that quantum groups will lead to a deeper understanding of the concept of symmetry in physics. On the physical side there are interrelations with the quantum inverse scattering method, the theory of integrable models, elementary particle physics, conformal and quantum field theories, and others. There are links with mathematical fields such as Lie groups, Lie algebras and their representations, special functions, knot theory, low-dimensional topology, operator algebras, noncommutative geometry, and combinatorics.

A striking feature of quantum group theory is the surprising connections with many, sometimes at first glance unrelated, branches of mathematics and physics. An algebraic approach to quantized coordinate algebras was given about this time by Yu.I. Woronowicz invented the quantum group SUq (2) and developed his theory of compact quantum matrix groups. These Hopf algebras will be called Drinfeld-Jimbo algebras in this book. Jimbo around 1985 of a class of Hopf algebras which can be considered as one-parameter deformations of universal enveloping algebras of semisimple complex Lie algebras. Its Hopf algebra structure was discovered later by E.K. Reshetikhin on the study of integrable XYZ models with highest spin. The algebra Uq (512 ) appeared first in 1981 in a paper by P.P. Faddeev and the Leningrad school on the inverse scattering method in order to solve integrable models. The birth of the new theory and its rapid development are results of a strong interrelation between mathematics and physics. The invention of quantum groups is one of the outstanding achievements of mathematical physics and mathematics in the late twentieth century. A number of topics and recent results from the more advanced general theory are developed and discussed. The basic quantum groups and quantum algebras and their representations are given in detail and accompanied by explicit formulas. The book is written with potential applications in physics and mathematics in mind. The authors cover a large but well-chosen variety of subjects from the theory of quantum groups (quantized universal enveloping algebras, quantized algebras of functions) and q-deformed algebras(q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus. It can also be used as a reference by more advanced readers. This book starts with an introduction to quantum groups for the beginner and continues as a textbook for graduate students in physics and mathematics.

Dxo viewpoint 2.5.7