Book Review

Book Review

The book under review was written by one of the principal developers of the subject. It is mainly based on the author's own research papers and contains many of his results, presented now in a self-contained form.

The book is not devoted to the standard quantum groups U_q(g), which are treated in greater detail in some other books (Chari and Pressley, Joseph, Lusztig, Kassel, Zhelobenko). Thus, the book does not contain crystal bases (discussed in Lusztig's monograph). And it does not cover the abstract deformation theory (for this, see the book of Shnider and Sternberg). It is, rather, a book about quantum groups themselves as mathematical objects considered from an original point of view. Their rigorous abstract algebraic treatment based on Hopf algebra techniques runs through the book as a kind of "backbone".

Thus, the author introduces and investigates dual quasitriangular Hopf algebras, which are, in some sense, a more fundamental notion than quantum enveloping algebras, since they can be associated to any solution of the quantum Yang-Baxter equation. Moreover, he defines the notions of twisting, *-structures, quantum-cocycle extensions of Hopf algebras, and bosonisation, and studies the corresponding structures. At the same time, the book contains a great many examples drawn from a broad range of mathematics and physics. One finds quantum groups in combinatorics and probability theory. There is a chapter devoted to bicrossproduct quantum groups associated to any group factorizing into subgroups. They are very different quantum groups from the usual U_q(g), but they are noncommutative and noncocommutative an d have remarkable self-duality properties as well. Majid relates them to Planck-scale physics.

Also, much research has been done in recent years on the kind of "quantum geometry" related to quantum groups. The author has an entire chapter devoted to his theory of Hopf algebras in braided categories or "braided groups". He gives a version of q-deformed linear algebra and differentiation on quantum planes A_q, which are braided groups with additive coproduct. This approach is applied to q-deformed physics, including q-Euclidean and q-Minkowski spaces and the q-Poincare group.

It is worth saying that there are interesting algebras which are not quantum groups or quantum planes, such as the Sklyanin algebra and the so-called Cherednik "reflection equation" algebra studied from a physical point of view in some papers of the Saint Petersburg school. In Majid's book we find that they can be given naturally the structure of braided groups. The author also describes a transmutation procedure suggested earlier in his original papers which converts quantum groups to braided ones and vice versa. (To avoid confusion we want to remark that the term "braided" is used in Kassel's book just for an ordinary quantum group possessing Drinfeld's quasitriangular structure (a quasitriangular Hopf algebra), whereas Majid uses this term for objects really "living" in a braided category, such as braided groups.)

In summary, this is a remarkable book, written in a rigorous style (all necessary statements are proved explicitly) by an acknowledged leader in the field. This book should be strongly recommended both to students and to professionals.

Reviewed by Dmitrii I. Gurevich

@ American Mathematical Society 1997 (reproduced by their kind permission)

Review of Foundations of Quantum Group Theory

From Math Reviews 97g:17016 17B37 18D99 81R50