##
**Introduction to quantum groups.**
*(English)*
Zbl 0788.17010

Progress in Mathematics (Boston, Mass.). 110. Boston, MA: Birkhäuser. xii, 341 p. (1993).

This book is about quantized enveloping algebras and their representations. These algebras – often called quantum groups – were introduced some 10 years ago by Drinfeld and Jimbo in connection with certain problems in statistical mechanics. However, as the author admits in the preface, he regards them here purely as a new development in Lie theory.

The book introduces quantum groups almost from scratch. No previous knowledge of semisimple Lie algebra theory is presupposed. Starting out with what he calls a Cartan datum \((I,\cdot)\) (in a way resembling how Kac-Moody algebras are often launched from a generalized Cartan matrix) the author begins by defining a \(\mathbb Q(v)\)-algebra f which will play the role of the positive (and negative) part of the quantum group \(U\) associated with \((I,\cdot)\). Instead of defining f via the quantum Serre relations he uses a bilinear form (the Drinfeld form) and sets f equal to the free \(\mathbb Q(v)\)-algebra on \(I\) modulo the radical of this form. Only much later in the book it is proved that the resulting quantum group is indeed the same as the one obtained by the usual generators and relations. Moreover, from the very beginning of the book he includes a treatment of \(\mathbb Z[v,v^{-1}]\)-forms of f and \(U\).

The above definitions are found in Part I of the book together with some of the basic tools (e.g. braid group operators) for studying these algebras and their modules. In addition this part contains the first results on integrable \(U\)-modules (like the complete reducibility theorem for certain such module categories).

In 1989 the author constructed his famous canonical basis for f (in the case of a symmetric Cartan datum of finite type). In fact, he gave two rather different constructions of this basis, one algebraic and one topological in nature. This basis plays a fundamental role in the book. Its topological construction relies on the theory of perverse sheaves and therefore this theory is reviewed in Part II. The theory is developed in such a way that it now applies to the case of an arbitrary Cartan datum. Moreover, this way of constructing the canonical basis has the advantage that it gives positivity results for the (multiplicative) structure constants for the basis. Actually, this last statement is (so far) only valid in the simply laced cases.

Part III contains Kashiwara’s approach to crystal bases. The author explores in a very neat way the interplays between Kashiwara’s construction and his own (two) construction(s).

In Part IV a modified \(\dot U\) of \(U\) is studied. The advantage of \(\dot U\) over \(U\) is that it is more appropriate for highest weight representations. It is shown here that \(\dot U\) has a canonical basis and applications are given for instance to coinvariants in tensor products. Moreover, \(\dot U\) has a natural \(\mathbb Z[v,v^{-1}]\)-form and therefore gives rise to analogous versions over any \(\mathbb Z[v,v^{-1}]\)-algebra. This is explored in Part V where several specializations of \(v\) are considered. It is here that we find the above mentioned relation between \(U\) (defined via the Drinfeld form) and the ‘usual’ quantum group (defined in terms of generators and relations). Also the quantum Frobenius homomorphism is defined and studied in this part.

In the final Part VI one finds more on the braid group actions. Braid group operators (on integrable U-modules) were introduced already in Part I but here they are studied in detail. In particular, it is shown that they do indeed satisfy the braid relations. Among the applications is a striking purely combinatorial character formula for the finite dimensional U-modules (in the simply laced case).

So this book is much more than an ‘introduction to quantum groups’. It contains a wealth of material. In addition to the many important results (of which several are new – at least in the generality presented here) there are plenty of useful calculations (commutator formulas, generalized quantum Serre relations, etc.).

The book introduces quantum groups almost from scratch. No previous knowledge of semisimple Lie algebra theory is presupposed. Starting out with what he calls a Cartan datum \((I,\cdot)\) (in a way resembling how Kac-Moody algebras are often launched from a generalized Cartan matrix) the author begins by defining a \(\mathbb Q(v)\)-algebra f which will play the role of the positive (and negative) part of the quantum group \(U\) associated with \((I,\cdot)\). Instead of defining f via the quantum Serre relations he uses a bilinear form (the Drinfeld form) and sets f equal to the free \(\mathbb Q(v)\)-algebra on \(I\) modulo the radical of this form. Only much later in the book it is proved that the resulting quantum group is indeed the same as the one obtained by the usual generators and relations. Moreover, from the very beginning of the book he includes a treatment of \(\mathbb Z[v,v^{-1}]\)-forms of f and \(U\).

The above definitions are found in Part I of the book together with some of the basic tools (e.g. braid group operators) for studying these algebras and their modules. In addition this part contains the first results on integrable \(U\)-modules (like the complete reducibility theorem for certain such module categories).

In 1989 the author constructed his famous canonical basis for f (in the case of a symmetric Cartan datum of finite type). In fact, he gave two rather different constructions of this basis, one algebraic and one topological in nature. This basis plays a fundamental role in the book. Its topological construction relies on the theory of perverse sheaves and therefore this theory is reviewed in Part II. The theory is developed in such a way that it now applies to the case of an arbitrary Cartan datum. Moreover, this way of constructing the canonical basis has the advantage that it gives positivity results for the (multiplicative) structure constants for the basis. Actually, this last statement is (so far) only valid in the simply laced cases.

Part III contains Kashiwara’s approach to crystal bases. The author explores in a very neat way the interplays between Kashiwara’s construction and his own (two) construction(s).

In Part IV a modified \(\dot U\) of \(U\) is studied. The advantage of \(\dot U\) over \(U\) is that it is more appropriate for highest weight representations. It is shown here that \(\dot U\) has a canonical basis and applications are given for instance to coinvariants in tensor products. Moreover, \(\dot U\) has a natural \(\mathbb Z[v,v^{-1}]\)-form and therefore gives rise to analogous versions over any \(\mathbb Z[v,v^{-1}]\)-algebra. This is explored in Part V where several specializations of \(v\) are considered. It is here that we find the above mentioned relation between \(U\) (defined via the Drinfeld form) and the ‘usual’ quantum group (defined in terms of generators and relations). Also the quantum Frobenius homomorphism is defined and studied in this part.

In the final Part VI one finds more on the braid group actions. Braid group operators (on integrable U-modules) were introduced already in Part I but here they are studied in detail. In particular, it is shown that they do indeed satisfy the braid relations. Among the applications is a striking purely combinatorial character formula for the finite dimensional U-modules (in the simply laced case).

So this book is much more than an ‘introduction to quantum groups’. It contains a wealth of material. In addition to the many important results (of which several are new – at least in the generality presented here) there are plenty of useful calculations (commutator formulas, generalized quantum Serre relations, etc.).

Reviewer: Henning Haahr Andersen (Aarhus)

### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

16T05 | Hopf algebras and their applications |

16T20 | Ring-theoretic aspects of quantum groups |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

20G05 | Representation theory for linear algebraic groups |