Tonny Springer is now in his eighties and since 1991 has been Emeritus at the University of Utrecht in the Netherlands (though still quite active: his “An extension of Bruhat’s lemma” appeared in J. Algebra as late as 2007). He has long been a major player in the field of algebraic groups. The book under review, currently in its second edition, is a mainstay in the literature. It is a source par excellence from which a beginner in linear algebraic groups can enter into this beautiful subject, under the tutelage of a master.
The present second edition (actually a reprint of the 1998 second edition) is an expanded version of Springer’s 1981 book of the same title, whose focus was on linear algebraic groups over an algebraically closed field. The present expanded edition also addresses the general case of an arbitrary ground field (in the latter seven chapters). Additionally, as the laudatory back-cover passages from Math. Reviews and Zentralblatt convey, Springer’s second edition contains treatments of a number of results not generally available in the literature at this introductory level, including sundry results by Rosenlicht, Borel, and Tits, as well as an elementary presentation of Tannaka’s theorem. Obviously the reader who works his way through these pages soon finds himself in the thick of things and is soon close to being able to have a go at research papers and research monographs.
In Linear Algebraic Groups Springer aims at a self-contained treatment of the subject in the title and he certainly succeeds; however, there is no denying that his compact treatment of, for instance, background from algebraic geometry and from Lie algebras would be more comfortably received by a reader with some experience in these areas. But Springer presents excellent (and numerous) good exercises to facilitate a rapid ascent for a neophyte, modulo serious commitment and a solid work ethic, so the adventurous student should simply dive in. Indeed, the first ten chapters of this edition of Linear Algebraic Groups largely correspond to the contents of Springer’s well-known and eminently successful 1981 first edition, based on the author’s 1978 lectures at Notre Dame, so that the reader is presented with a very solid broad-based introduction (a true first course in itself!) before the aforementioned case of arbitrary base fields is broached.
Finally, in addition to the abundance of good exercises already mentioned, each chapter comes equipped with an endnote for a bit of history and context, as well as indications of where to go next. And all of it is done in a very clear style, making for a smooth and readable presentation. Thus, Linear Algebraic Groups is a superb choice for any one wishing to learn the subject and go deeply into it quickly and effectively.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.