This small (141 pages) book is designed as an introduction to quantum groups and related topics. It is an informally written guide with many details not given. The ultimate emphasis is on categorical ideas and diagrams. We find it useful to list the chapter titles. 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.
The only quantum (matrix) groups discussed are \(M_q(n)\) and \(\text{GL}_q(n)\). In particular, there is no mention of quantum universal enveloping algebras. Ch. 1 defines monoids, groups and rings and their morphisms, as well as categories with relevant diagrams. This might suggest that the book could be read by beginners. But a previous knowledge of algebra and category theory would be almost necessary (more of this later, re its possible use as a text). Ch. 2 gives some philosophical remarks preparing the reader to think geometrically. For example, a morphism of algebras from \(A\) to \(B\) is called a \(B\)-point of \(A\).
Ch. 3 introduces \(M_q(2)\) and \(\text{GL}_q(2)\) as algebras. The properties needed for the constructions are proved by introducing the quantum plane and quantum superplane (exterior algebra). While the latter is used to sketch a proof of the multiplicativity of \(\det_q\), there is no conceptual proof given of its centrality (a direct proof is assigned as an exercise). Ch. 4 is fairly standard, emphasizing \(R\)-\(S\) bimodules (\(R,S\) rings) as pictures \(M\colon R\to S\). In Ch. 5, \(M\colon R\to S\) gives \(M^*=\operatorname{Hom}_R(M,R)\colon S\to R\). \(M\) is called Cauchy if the usual map of \(M^*\otimes L\) to \(\operatorname{Hom}_R(M,L)\) is an isomorphism for all \(L\). This is equivalent to \(M\) being finitely generated projective as left \(R\)-module, but other equivalent conditions are given (this is called the fundamental theorem of Morita theory) indicating the categorical nature of the condition. Ch. 6 is standard. Ch. 7 is also somewhat standard, but starts to stress diagrams. Elements often called group-like are here called set-like. The author defines primitive elements in a coalgebra, but this makes no sense in general. The examples given are in bialgebras, which have a unit element 1.
Ch. 8 constructs the dual coalgebra \(A^0\) of an algebra \(A\) over a ring \(R\) in a way designed to apply for an algebra over an arbitrary ring \(R\). However, the subsequent development is for \(R\) a field. An ideal \(I\) of \(A\) is called coCauchy if \(A/I\) is Cauchy (as \(R\)-module). Then \(A^0\) is defined as those elements of \(A^*\) vanishing on a coCauchy ideal of \(A\). When \(R\) is a field, the proofs given in the construction suggest \(I\) is coCauchy if and only if \(I\) is cofinite-dimensional, so that \(A^0\) is the usual one.
Ch. 9 gives the descriptions of the commutative algebras \(M(n)\) and \(\text{GL}(n)\) as bialgebra and Hopf algebra, respectively. The construction sketched omits the properties of the ideals being factored out of free algebras needed to be compatible with the costructure and the antipode. These properties, e.g., the idea of a coideal, are assigned as exercises at the end of the chapter. The quantum versions \(M_q(n)\) and \(\text{GL}_q(n)\) are described, but no details are given. The justification will come at the end of the book in Ch. 18, after the relevant categorical framework has been established.
Ch. 10 goes via comodules, is highly categorical, and obtains the categorical equivalence of \(\text{Com}_R(C)\) and \(\text{Mod}_R(C^*)\) when \(C\) is Cauchy (as an \(R\)-module). This is done by the fundamental theorem of Morita theory from Ch. 5. When \(M\) is Cauchy, an important role is played by a coalgebra structure on \(M\otimes M^*\).
Starting with Ch. 11, the point of view becomes more and more categorical and pictorial. Ch. 11 introduces monoidal categories, braid groups and quasitriangular bialgebras. The discussion of Tannaka duality in Ch. 16 is completely categorical.
The book has some exercises, and a final Chapter 19 gives solutions (or indications of solutions). This suggests the possible use of the book as a text. In the Introduction, the author indicates that the set theoretic approach to the study of algebra should be replaced by a more categorical approach. It is easy to agree with this, given all the current ideas in non-commutative geometry. The author has taught a course at his university based on notes which developed into the book. The book is very well written. It is quite concise, so that a student would need considerable guidance in filling in all the details. So it would be a good idea to have a solid background in traditional algebra as well as in category theory.