Quantum groups from coalgebras to Drinfeld algebras. A guided tour.

*(English)*Zbl 0845.17015
Cambridge, MA: International Press. 496 p. (1993).

Although there is still no generally accepted definition of the notion of a quantum group, most workers in the field would agree that quantum groups are Hopf algebras (or perhaps quasi-Hopf algebras in the sense of Drinfeld – see below), and that the most interesting examples are deformations of universal enveloping algebras of Lie algebras or of algebras of functions on groups. Thus, the deformation theory of Hopf algebras plays a central role in the theory of quantum groups. This book is the first detailed treatment of the deformation theory of Hopf algebras and its applications.

“The first three chapters form an introduction to bialgebras and Hopf algebras. Influenced by Majid, we adopt the categorical point of view (including reconstruction) early on. Chapter 3 ends with two gems: Ree’s shuffle algebra which has applications to computer science in the abstract theory of languages, and Chen’s iterated integrals.”

After a digression in Chapter 4 to discuss the Hopf algebra approach to the representation theory of symmetric groups, Chapters 5 and 6 are devoted to the theory of braided categories, including a treatment of MacLane’s coherence theorems via Stasheff polyhedra.

“Chapter 7 is a mini course on statistical mechanics on a two-dimensional lattice. It is self contained, but only goes as far as motivating the star triangle relation and the Yang-Baxter equation.”

Chapter 8 introduces quasi-Hopf algebras, here called Drinfeld algebras. A quasi-Hopf algebra is a generalization of the notion of a Hopf algebra in which the comultiplication is only required to be coassociative up to conjugation.

The homological tools necessary for the deformation theory of (quasi-) Hopf algebras are treated in detail in Chapters 9, 10 and 13. Applications of the theory to deformations of universal enveloping algebras are given in Chapter 11. Although most of the results in this chapter are due to Drinfeld, this is the first time many of them have received detailed proofs in print.

Chapter 12 deals with Drinfeld’s reinterpretation and proof of Kohno’s theorem on the monodromy of the Knizhnik-Zamolodchikov equations. It was this work which motivated Drinfeld’s introduction of the notion of a quasi-Hopf algebra.

“The first three chapters form an introduction to bialgebras and Hopf algebras. Influenced by Majid, we adopt the categorical point of view (including reconstruction) early on. Chapter 3 ends with two gems: Ree’s shuffle algebra which has applications to computer science in the abstract theory of languages, and Chen’s iterated integrals.”

After a digression in Chapter 4 to discuss the Hopf algebra approach to the representation theory of symmetric groups, Chapters 5 and 6 are devoted to the theory of braided categories, including a treatment of MacLane’s coherence theorems via Stasheff polyhedra.

“Chapter 7 is a mini course on statistical mechanics on a two-dimensional lattice. It is self contained, but only goes as far as motivating the star triangle relation and the Yang-Baxter equation.”

Chapter 8 introduces quasi-Hopf algebras, here called Drinfeld algebras. A quasi-Hopf algebra is a generalization of the notion of a Hopf algebra in which the comultiplication is only required to be coassociative up to conjugation.

The homological tools necessary for the deformation theory of (quasi-) Hopf algebras are treated in detail in Chapters 9, 10 and 13. Applications of the theory to deformations of universal enveloping algebras are given in Chapter 11. Although most of the results in this chapter are due to Drinfeld, this is the first time many of them have received detailed proofs in print.

Chapter 12 deals with Drinfeld’s reinterpretation and proof of Kohno’s theorem on the monodromy of the Knizhnik-Zamolodchikov equations. It was this work which motivated Drinfeld’s introduction of the notion of a quasi-Hopf algebra.

Reviewer: A.N.Pressley (London)

##### MSC:

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

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

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

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |

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