A quantum groups primer (Q2784582)

The history of the development of the theory of Hopf algebras can be roughly divided into two phases. The first phase began with their introduction in 1940s by H.\ Hopf and culminated in the late 1960s with the seminal paper by Milnor and Moore and the monograph by Sweedler. At this time Hopf algebras were mainly studied by algebraists and their main applications were in algebraic topology. The second phase of the development began with V.\ Drinfel'd's address to the International Congress of Mathematicians in 1986 and a seminal paper by M.\ Jimbo. This coincided with the beginnings of noncommutative geometry. Drinfel'd and Jimbo constructed new examples of Hopf algebras arising from deformation or ``quantisation'' of enveloping Lie algebras. Thus Hopf algebras became popular as quantum groups and found numerous applications mainly in theoretical physics but also in various branches of pure mathematics such as noncommutative geometry, knot theory, category theory, representation theory, etc. Since the 1990s a number of monographs and textbooks devoted to quantum groups and, more generally, Hopf algebras have been published. Among these books, \textit{S. Majid}'s [Foundations of quantum group theory, Cambridge University Press (1995; Zbl 0857.17009)] stands out as the, probably, most wide-ranging description of the theory and applications of Hopf algebras that appeals to both theoretical physicists and pure mathematicians. This monograph is an excellent reference (and often a true ``eye-opener'') for researchers working in quantum groups. However, the sheer size of this monograph makes it rather hard to adapt as a basis of a one-term post-graduate lecture course. Fortunately, this is no longer necessary.NEWLINENEWLINEIn the Spring of 1998, S.\ Majid gave a one-term post-graduate lecture course at Cambridge (or a Part III of Mathematical Tripos course in the University of Cambridge terminology) devoted to quantum groups. The lecture notes to this course are now published as the book under review or \textit{A Quantum Group Primer}. The division of the book into 24 sections follows closely the original format of the lecture course consisting of 24 one-hour lectures. The wide range of topics covered within this rather limited space is truly amazing. The book begins by describing basic algebraic structures underlying quantum groups (algebras, coalgebras, bialgebras, comodules etc.). Very quickly the reader is led to examples and to more advanced algebraic topics such as quasitriangular structures and quantised universal enveloping algebras at roots of unity. The second part of the book deals with representation theory of quantum groups in terms of braided categories. We learn, among others, about crossed modules, \(q\)-Hecke algebras, rigid objects, duality, quantum dimension, then about braided Hopf algebras and their differential geometry (braided differentiation). In this part quantised universal algebras are derived by an inductive braided-categorical procedure. The last part of the book describes various more advanced topics, motivated by the role of quantum groups as symmetries of noncommutative spaces. In particular the topics include Lie bialgebras, Poisson geometry, connections on quantum principal bundles, including an explicit example of the \(q\)-deformed Hopf fibration and the Dirac \(q\)-monopole example.NEWLINENEWLINES.\ Majid is well-known for his lively and very informative style of writing, and the reviewed book confirms this opinion. Thus the book is very well written, the proofs contain enough details to make them easily readable but still challenging enough to keep students interested. The material is presented in a concise and economical way, stressing main ideas rather than technical details. The reader interested in such details can find them in the author's \textit{Foundations of Quantum Group Theory} or in original papers. The book is supplemented with a selection of problems from the original lecture course.NEWLINENEWLINEIn the preface the author writes: ``This is a self-contained first introduction to quantum groups as algebraic objects. [\dots] The approach is basically that taken in my 1995 textbook, to which the present work can be viewed as a companion `primer' for pure mathematicians. As such it should be a useful complement to that much longer text''. The reviewer is convinced that this book fulfils all these aims. I can full-heartily recommend this work as a basis for a one-term postgraduate course or as an introductory text to all mathematicians who would like to learn quickly the main ideas, techniques and wide range of applications of quantum group theory.