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Quantum Groups: A Path to Current Algebra

Ross Street
Cambridge University Press
Publication Date: 
Number of Pages: 
Australian Mathematical Society Lecture Series 19
[Reviewed by
Fabio Mainardi
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The ideal readers for this book are graduate students and researchers interested to an overview of the theory of quantum groups. Some acquaintance with algebraic groups would be helpful. There is a fair amount of category theory throughout the text, and it couldn’t be otherwise; unfortunately, it is not always easy, for the beginner, to see the motivations behind some constructions.

Some good features of this book:

  • Exercises are copious, with solutions at the end of the book
  • Examples are abundant
  • The chapters are never lengthy

Some branches of mathematics or physics related to quantum groups: Yang-Baxter equation, quantum inverse scattering method, knots, braid group.

The author begins with a review of the basic algebraic structures (groups, monoids, etc.) from the point of view of category theory. The general linear group is then re-defined according to this philosophy; this is really essential in order to understand the definition of the quantum general linear group (chapter 3).

Actually, there is no single, all-encompassing definition of quantum group (as far as I know). However, the general philosophy behind this theory is the following: according to A. Connes point of view, to a topological group we can associate its commutative algebra of continuous functions and, reciprocally, the group is determined up to isomorphism by that algebra. Now, imagine you want to extend this correspondence to a more general setting, where you allow certain non-commutative ‘deformations’ of the function algebras. Then, the corresponding abstract objects are what one calls ‘quantum groups’. So a quantum group is, roughly speaking, a non-commutative algebra with extra structure (in fact, a Hopf algebra with additional structure). The terminology is therefore a bit confusing, since a quantum group is not really a group…

This general picture can be very well (and easily) understood in the case of the quantum general linear group, so the reader should perhaps spend some time on the first three chapters. The remaining chapters are of a more technical nature.

A curious and pleasant thing about this book: the (Australian) author has reproduced at the end of some chapters the silhouettes of certain ancient Australian rock paintings, located in Bradshaws (, a magnificent example of Australian aboriginal art.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at

Introduction; 1. Revision of basic structures; 2. Duality between geometry and algebra; 3. The quantum general linear group; 4. Modules and tensor products; 5. Cauchy modules; 6. Algebras; 7. Coalgebras and bialgebras; 8. Dual coalgebras of algebras; 9. Hopf algebras; 10. Representations of quantum groups; 11. Tensor categories; 12. Internal homs and duals; 13. Tensor functors and Yang-Baxter operators; 14. A tortile Yang-Baxter operator for each finite-dimensional vector space; 15. Monoids in tensor categories; 16. Tannaka duality; 17. Adjoining an antipode to a bialgebra; 18. The quantum general linear group again; 19. Solutions to exercises; References; Index.