The book under review is intended, on the one hand, to be used by researches in areas related to Hopf algebras, reflecting the influence of quantum groups in the field and recent advances in the classification of pointed Hopf algebras. As pointed out by the author in the introduction, the contents of the book do not cover (and are neither covered by) the material in other texts on the subject, like the books byE. Abe [Hopf algebras. Cambridge Tracts in Mathematics 74. Cambridge: Cambridge University Press (1980;Zbl 0476.16008)];S. Dăscălescu, C. Năstăsescu andŞ. Raianu [Hopf algebras. An introduction. Pure and Applied Mathematics 235. New York: Marcel Dekker (2001;Zbl 0962.16026)];S. Montgomery [Hopf algebras and their actions on rings. Regional Conference Series in Mathematics 82. Providence: AMS (1993;Zbl 0793.16029)] andM. E. Sweedler [Hopf Algebras. New York: W. A. Benjamin (1969;Zbl 0194.32901)]. Besides from this, the book presents an excellent introduction to the theory, suitable for a graduate course on the subject. No previous knowledge on Hopf algebras is assumed, having as only prerequisites basic notions of elementary abstract algebra, linear algebra and rudiments of category theory. A big number of exercises of different level of difficulty are proposed along the text, which include in particular special features or applications to a variety of concrete examples, further results and categorical aspects of the corresponding material. Interesting and up-to-date historical and bibliographical comments are provided at the end of each of the sixteen chapters.
The contents of the book can be roughly summarized as follows. After a first chapter devoted to recalling some notions of linear algebra and topological aspects related to duality, the book starts with the study of coalgebras over a field and their categories of comodules, including a study of the coradical filtration.
Bialgebras and Hopf algebras are introduced next. The remaining part of the book is devoted to the development of different fundamental aspects of Hopf algebras. They are introduced in Chapter 7, where some families of nontrivial examples involving \(q\)-commutation relations are presented. This chapter contains also some important constructions related to cocycles, skew pairings and twists, a study of Hopf algebra filtrations, the cofree Hopf algebra on an algebra and the shuffle algebra.
The next chapters contain the structure theorem of Hopf modules of Larson and Sweedler, the Nichols-Zoeller freeness theorem. Chapter 10 presents a treatment of the important notion of an integral in a Hopf algebra and its applications to the structure of a finite-dimensional Hopf algebra.
Chapter 11 gives an introduction to monoidal and braided categories and presents several constructions related to Hopf algebras. In particular, it discusses module and comodule algebras, algebras and coalgebras, Yetter-Drinfeld (braided) Hopf algebras and the biproduct (also called bosonization) construction.
The next chapters develop the notion and structure of a quasitriangular Hopf algebra, the Drinfel’d double constructions and the dual notion of a coquasitriangular Hopf algebra. Chapter 15 contains results on pointed Hopf algebras and related structures and constructions which are meant to set stage for their study and classification. The final chapter discusses finite-dimensional Hopf algebras in characteristic 0 with emphasis in classification results.

Reviewer: Sonia Natale (Córdoba)