An Invitation to Quantum Groups and Duality: From Hopf Algebras to Multiplicative Unitaries and Beyond

Thomas Timmermann was writing a dissertation. He was to delve into the depths of a whole research field and learn all the basic constructions and ideas along with a good amount of background material in neighboring fields before zeroing in on a specific research problem. As many others do, he probably did this by reading books and then moving on to articles and trying to squeeze the juice out of many many pages of information that were scattered across the vast universe known as Mathematical Literature. Then he probably sat down and just did the math. And he completed what he had set out to do: he got his degree in 2005.

He could have moved on; he could have simply put all of that background work aside and just continued with his research agenda. Timmermann, however, chose to do something else. He decided to write a book which would collect together the bits and pieces he had managed to find out about during his readings, a book which would be the result of many hours searching the literature for the main constructions, main ideas, main motivational examples. Timmermann decided he wanted to write the book he would have wanted to read when he started to work on his dissertation.

The book reviewed here is the end result of this effort. It is very well written, and it is a pleasure to read. It fills a noticeable gap in the literature, and the extensive details included help its intended audience follow the constructions and deductions with ease.

Hopf algebras were first introduced by Hopf in the 1940s. They were studied extensively, in the following decades, at the beginning purely algebraically. Powerful connections to the representation theory of algebraic groups and Lie algebras were studied in the late 1970s.

In the early 1980s, just when one could have assumed that they would perhaps go out of fashion as abstract algebraic structures which are of interest only to a few specialists, Drinfeld proposed them as the ultimate framework for understanding quantum groups. This revolutionized the study of Hopf algebras by introducing a large collection of examples, bringing up interesting questions, and relating them to mathematical physics. Today the field is still thriving; there are many articles and research monographs on various aspects of Hopf algebras and quantum groups, and several textbooks which introduce the topic to the modern graduate student.

The book under review fits into the literature as follows: Even though the last few chapters focus on the author's own dissertation research and read somewhat more like a monograph, the book as a whole is first and foremost a book written for beginners in the field. Chapter 1 introduces the basic notion of Hopf algebras in a way accessible to any graduate student in mathematics or anyone with an equivalent background. This chapter is very generous with all kinds of motivational tidbits, and details of basic constructions which are not always easy to find in the more modern texts on Hopf algebras and quantum groups. For instance modern texts will generally leave their readers to prove for themselves that the antipode of a Hopf algebra is unique. This is rather straight-forward but the beginner will benefit from seeing the explicit computations, which are provided on p.16 of Timmermann's book.

The notion of duality, which is central to the rest of the text, is first introduced in Chapter 1, and then developed further in Chapter 2. The basic issue is that the standard (linear) dual construction when applied to a given Hopf algebra does not always yield a Hopf algebra, and furthermore, that the double dual is not always isomorphic to the original. There are several ways of getting around this issue, of course. For example one can restrict the kinds of elements one allows into the dual, thus redefining what the dual should mean (i.e., the restricted dual). Timmermann follows a different path and instead focuses on the notion of a multiplier Hopf algebra, a Hopf-like structure on an algebra A whose comultiplication maps to the multiplier algebra of A⊗A. This proves to be quite productive. He can then go on to define duals and show that they have desirable properties.

In the rest of the book we learn many facts and constructions related to the theory of quantum groups as studied by operator algebraists. A C^* structure is added to the Hopf algebra structure, and the quantum groups that are described and studied are assumed to mostly be function algebras and their generalizations. Due to this focus, the dual Lie-theoretic point of view, which relates quantum group theory to the universal enveloping algebras of Lie algebras and their deformations, is not as emphasized here as in some other modern texts. Given the extent of the primary research literature that does indeed take the operator algebraists' point of view, this is clearly not a deficiency but a welcome trait. A brief appendix on the basics of C^*-algebra theory and an extensive bibliography are provided for reference.

Overall the reviewer hopes and expects that this book will find its (sizeable and grateful) audience; it is an excellent introduction and its writing was definitely time well-spent by its author, no matter what the standard advice for young mathematicians about writing books is.

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Gizem Karaali is assistant professor of mathematics at Pomona College. In the world of quantum groups, she resides more often on the Lie-theoretic side but is slowly figuring her way around in the neighboring countries.