Hopf algebras are abstractions of algebras of functions on a group. As such, they were introduced to topology in the first half of the last century. Roughly speaking, Hopf algebra is an associative algebra that also carries a “comultiplication”, and these two structures are compatible in an appropriate sense. The second advent of Hopf algebras was marked by a seminal talk on quantum groups, presented by V. Drinfel’d in Berkeley in 1986. Since then, Hopf algebras have found a great number of applications, not only in the theory of quantum groups (which are particular types of Hopf algebras) but also in low-dimensional topology, algebra and geometry. The book under review is a collection of expository articles that attempt to summarize the recent developments and offer a new understanding of classical topics. The contributions were written by N. Andruskiewitsch, S. Gelaki, G. Letzler, A. Masuoka, D. Nikshych, F. van Oystaeyen, D. Radford, P. Schauenburg, H.-J. Schneider, M. Takeuchi, L. Vainerman and Y. Zhang. Most of the authors were participants of the Hopf Algebras Workshop held at MSRI as a part of the 1999-2000 Year on Noncommutative Algebra.

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