An *affine algebraic group*, say over an algebraically closed field, is a representable functor from the category of commutative algebras over the given field to the category of groups, where the representing object (the coordinate ring of the algebraic group) is a commutative Hopf algebra over the field *k*. Moreover, there is an (anti) equivalence between the category of *affine groups* over *k* and the category of *Hopf algebras* over *k*. This view of the familiar matrix groups emphasizes Grothendieck’s philosophy that mathematical objects are characterized by certain functions on them.

In the 1980s and motivated by some mathematical physics problems related to the so-called quantum Yang-Baxter equations, it was found that deformations of the universal enveloping algebra of a semisimple Lie algebra could give some solutions for the problems being studied. Since a complex semisimple Lie algebra is the Lie algebra of a semisimple complex Lie group, by the perfect Hopf pairing between the coordinate ring of the given group and the enveloping algebra of the Lie algebra associated to the given group, deformations of the universal enveloping algebra of the Lie algebra correspond to deformations of the Hopf algebra of regular functions on the Lie group.

Following Grothendieck’s philosophy, Drinfeld proposed to study these deformed Hopf algebras as if they were the algebras of regular functions on *a not yet existing object*, a quantum deformed group. Thus, *quantum groups, *a cleverly chosen name and a beautiful analogy, came into the mathematical landscape.

Since representations of an algebraic group correspond to modules over the corresponding Hopf algebra, one is lead to study the representations of the quantum deformed Hopf algebras (quantum groups). Kashiwara (1990: *Crystallizing the q-analogue of universal enveloping algebras*. Duke Math. J. 73, 383-413) showed that these modules have a canonical or *crystal* basis, equipped with a semi-simple action of the generators of the quantum group (actually, the quantum algebra).

Crystal bases also appeared in the work of Littelmann on bases (indexed by Young tableaux) of modules of sections of line bundles on flag varieties, elaborating on previous work of Lakshmibai and Seshadri on standard monomial theory.

The book under review, following ideas of Kashiwara and Stembridge, develops an axiomatic combinatorial approach to the construction of crystal bases associated to finite dimensional quantum-deformed Lie algebras. The cue for this approach comes from the rich combinatorics of Young tableaux, for example in the construction of the irreducible representations of the symmetric and general linear groups by means of certain Young tableaux.

This combinatorial approach is by no means an easy one, but compared with the deep methods on quantum group theory used by Kashiwara and Lusztig, or Littelmann, it has the feeling of being a bit more friendly, keeping the constructions grounded by comparing them with the combinatorial approach to the representation theory of the symmetric and general linear groups.

The introduction to crystal bases given in this book is accessible to graduate students and researchers. Its combinatorial approach is reader-friendly and invites the reader to get involved, either working on the exercises listed at the end of each chapter or by playing with the computational implementation on freely-available software, sage in this case, where the authors have many built-in accessible examples and some algorithms for specific crystal bases.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is [email protected].