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Compact Quantum Groups and Their Representation Categories

Sergey Neshveyev and Lars Tuset
Société Mathématique de France
Publication Date: 
Number of Pages: 
Cours Spécialisés 20
[Reviewed by
Michael Berg
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The authors start this austere book (albeit an introductory treatment) with a particularly provocative comment: “The term ‘quantum group’ was popularized in the 1980s and … does not have a precise meaning.” One infers therefore that this young subject is still in a state of growth, or perhaps mathematical adolescence, for its antecedents are found in the theory of Hopf algebras, dating to the 1950s — so maybe it’s a young adult.

To give a flavor of what’s going on here, Hopf observed almost a decade before, perhaps prophetically, that a compact group G possesses the property that its cohomology ring \(H^*(G)\) admits a homomorphism into \(H^*(G)\otimes H^*(G)\). The authors immediately go on to observe that on a related note, and more elementarily, if \(G\) is a finite group then the algebra \(C(G)\) of functions on \(G\) admits a kindred homomorphism from \(C(G)\) to \(C(G)\otimes C(G)\), and “[w]hat is important is that the pair [consisting of the algebra \(C(G)\) and this homomorphism] contains complete information about … \(G\) … including [the wherewithal] to recover the group law.”

One then goes on to couple this with a theme due to G. I. Kac, namely a generalization of Pontryagin duality involving an exploitation of von Neumann algebras and in this way we get the theory of Kac algebras, and we’re within epsilon (or maybe \(3\varepsilon\)) of our goal: in the middle 1980s Jimbo and Drinfeld “introduced new Hopf algebras by deforming universal enveloping algebras of semisimple Lie groups … Drinfeld also introduced their dual objects … [and] suggested the term ‘quantum groups’ for Hopf algebras related to these constructions.” Subsequently, and most relevant to the present review, the authors, Neshveyev and Tuset, state that they “take the analytic point of view, meaning that [they] work with algebras of, preferably bounded, operators on Hilbert spaces,” and that “[t]he goal of this short book is to introduce the reader to this beautiful area of mathematics.”

Accordingly the book is split into four chapters, totaling about 150 pages (sans end matter), and although the first two chapters certainly present “a general theory of compact quantum groups,” in itself a highly meritorious act, they put serious demands on the uninitiated reader even at the indicated comparatively foundational level. But the authors are happily rather specific in what this reader should have under his belt: “a basic course in \(C^*\)-algebras … a minimal knowledge of semisimple Lie groups … [and it would be] beneficial to have some acquaintance with category theory.” And it is of course the case that category theory is like chocolate: if you only have an acquaintance with it you don’t really have an acquaintance with it — and just look at the title of the second chapter: “\(C^*\)-Tensor Categories.” The rubber hits the road — hard — right away (we’re only on p. 35).

So, indeed it is on target that this compact book (if you’ll excuse a cheap play on words) is published in the Cours Spécialisés series of the French Mathematical Society: it’s genuinely specialized stuff. But it’s also very pretty and very exciting given that it has connections to such a wonderful arrangement of subjects, from algebraic topology and the theory of locally compact groups to functional analysis, Lie groups, and category theory.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

  • Compact quantum groups
  • \(C^*\)-tensor categories
  • Cohomology of quantum groups
  • Drinfeld twists
  • Bibliography
  • List of symbols
  • Index