Preface This volume is an outgrowth of the Colloquium on Quantum Groups and Hopf Algebras which took place at the Hotel del Lago, near the city of La Falda, in Sier- ras de Cordoba, Argentina from August 9 to August 13, 1999. The Colloquium, sponsored by the Facultad de Matematica, Astronomia y Fisica, Universidad Na- cional de Cordoba, was attended by 46 participants from 13 countries, including several graduate and undergraduate students. There were two introductory courses as well as 29 conferences--see the program on page xiii. One purpose of the meeting was to bring together researchers to discuss recent developments in Hopf algebra theory, one of the most important of which is the influence of quantum groups. The second purpose was to stimulate further devel- opment of the subject by South American mathematicians, especially by those in Cordoba and Montevideo. This development is promoted by the first two-named Editors with fundamental help of the third. The articles which follow represent most of the subjects covered in the Collo- quium. Some of the material presented here consists of expanded versions of the talks given at the meeting. Other articles were written especially for the present volume. Some were kindly prepared by the authors following suggestions of the Ed- itors in order to present an overview, survey or introduction to subjects of current interest lacking a systematic presentation in the existing literature. We describe the contents of this volume in some detail. There are papers on general structure of Hopf algebras. A survey is given by Cohen on characters on Hopf algebras and generalizations. Kauffman and Radford present an alternative proof of the existence of non-zero integrals in finite dimensional Hopf algebras. A finite dimensional Hopf algebra is Frobenius because of the existence of inte- grals how far one could develop the theory in this direction, without assuming the Hopf condition, is investigated in the article "BiFrobenius algebras", by Doi and Takeuchi. Other articles are concerned with classification of semisimple and pointed Hopf algebras Masuoka's contribution deals with the semisimple case and the papers by Grana, and Milinski and Schneider deal with the pointed case. Braided Hopf algebras play a prominent role in classification problems the survey of Takeuchi covers the basic theory of these objects. On the other hand, Grana relates a class of pointed Hopf algebras with set-theoretic solutions of the quantum Yang-Baxter equation and this is the topic of the paper by Lu, Yan and Zhu. The paper by Westreich surveys inner and outer actions, and Galois theory. The papers by Garcia and Trinchero, and by Sachse, are devoted to *-Hopf algebras and have a more physical flavor. It is worth noticing that besides what could now-a-days be considered as "classical" subjects within the theory, addressed ix

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