This book adds to the growing collection of literature on quantum groups. Some of the other books are by V. Chari and A. Pressley, J. Fuchs, A. Joseph, C. Kassel, G. Lusztig, Yu. I. Manin, S. Montgomery, R. Shnider and S. Sternberg, and V. Turaev. The emphasis ranges from physical (as in Fuchs) to the purely algebraic (as in Montgomery). The book under review is a revision and expansion of notes for a course given in 1994 by the author. It begins with a thorough treatment of \(U_q ({\mathfrak {sl}}_2)\) and its representations, which serves to introduce the basic ideas of the subject, as well as being useful later in the more general case of \(U_q ({\mathfrak g})\) and its representations, where \({\mathfrak g}\) is a complex (finite-dimensional) semisimple Lie algebra. The base field for \(U_q ({\mathfrak g})\) is essentially arbitrary (of characteristic not two). For \(U_q ({\mathfrak {sl}}_2)\), \(q\) is arbitrary, the representation theory distinguishing between \(q\) a root of unity or not. In the general case, the author only gives complete details for \(q\) a root of unity. In contrast, Kassel only gives a complete treatment for \(U_q ({\mathfrak {sl}}_2)\). The last three chapters of Jantzen’s book deal with crystal bases, whereas Kassel has extensive applications to knot theory and to monodromy of Knizhnik-Zamolodchikov equations. Both Jantzen and Kassel can nicely serve as a textbook for a first course on quantum groups.
Jantzen’s book is carefully written, yet there is an agreeable, sometimes informal, spirit with which he tries to indicate to the reader what is really happening. One pleasant feature is his placing of long computations in appendices at the end of certain chapters.
While \(R\)-matrices and \(k_q [G]\), the quantum version of functions on the algebraic group \(G\), are mentioned in Chapter 7, the emphasis is on the universal enveloping algebra. The author has struck a reasonable balance in trying to make his book accessibe to students entering the study of quantum groups.
We list the titles of the Chapters 0-11. Ch. 0. Gaussian binomial coefficients, Ch. 1. The quantized enveloping algebra \(U_q ({\mathfrak {sl}}_2)\), Ch. 2. Representations of \(U_q ({\mathfrak {sl}}_2)\), Ch. 3. Tensor products or: \(U_q ({\mathfrak {sl}}_2)\) as a Hopf algebra, Ch. 4. The quantized enveloping algebra \(U_q ({\mathfrak g})\), Ch. 5. Representations of \(U_q ({\mathfrak g})\), Ch. 5A. Examples of representations, Ch. 6. The center and bilinear forms, Ch. 7. \(R\)-matrices and \(k_q [G]\), Ch. 8. Braid group actions and PBW type basis, Ch. 8A. Proof of proposition 8.28, Ch. 9. Crystal bases I, Ch. 10. Crystal bases II, Ch. 11. Crystal bases III.

Reviewer: E.J.Taft (New Brunswick)