Braid groups. With the graphical assistance of Olivier Dodane.

*(English)*Zbl 1208.20041
Graduate Texts in Mathematics 247. New York, NY: Springer (ISBN 978-0-387-33841-5/hbk). xii, 340 p. (2008).

Braid groups were formally introduced by Emil Artin in the 1920s to formalize topological objects modelling the intertwining of several strings in Euclidean 3-space, though they appeared in disguised form in a paper of Adolf Hurwitz in 1891. In 1983 V. F. R. Jones [Invent. Math. 72, 1-25 (1983; Zbl 0508.46040)] discovered new representations of braid groups, from which he derived his celebrated polynomial of knots and links. Some remarkable recent results in the field are P. Dehornoy [Trans. Am. Math. Soc. 345, No. 1, 115-150 (1994; Zbl 0837.20048)], D. Krammer [Invent. Math. 142, No. 3, 451-486 (2000; Zbl 0988.20023) and Ann. Math. (2) 155, No. 1, 131-156 (2002; Zbl 1020.20025)].

This book is a comprehensive introduction to the theory of braid groups. Assuming only a basic knowledge of topology and algebra, it is intended mainly for graduate and postdoctoral students. The book consists of seven chapters together with four short appendices on \(\mathrm{PSL}_2(\mathbb{Z})\), fibrations and homotopy sequences, Birman-Murakami-Wenzl algebras, and left self-distributive sets in order. Chapter 1 is concerned with the foundations of the theory of braids and braid groups. The central result of Chapter 2 is the Alexander-Markov description of oriented links in terms of Markov equivalence classes of braids. Chapter 3 is devoted to the Burau representation [introduced in W. Burau, Abh. Math. Semin. Hamb. Univ. 11, 179-186 (1935; Zbl 0011.17801)] and the Lawrence-Krammer-Bigelow representation [introduced in R. J. Lawrence, Commun. Math. Phys. 135, No. 1, 141-191 (1990; Zbl 0716.20022)]. Chapter 4 is concerned with the symmetry groups and the Iwahori-Hecke algebras. Chapter 5 deals with a classification of the finite-dimensional representations of the generic Iwahori-Hecke algebras in terms of Young diagrams. Chapter 6 is devoted to the Garside solution of the conjugacy problem in the braid groups. Chapter 7 presents the orderability of the braid groups, proving that the braid group \(B_n\) is orderable for every \(n\).

It is impossible to cover all the facets of the theory of braid groups in such a book of reasonable volume as this book. One important area entirely untouched is the connections with mathematical physics, quantum groups, Hopf algebras and braided monoidal categories, for which the reader is referred to G. Lusztig, [Introduction to quantum groups. Boston: Birkhäuser (1993; Zbl 0788.17010)], V. Chari and A. Pressley, [A guide to quantum groups. Cambridge: Cambridge University Press (1995; Zbl 0839.17010)], V. G. Turaev, [Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics 18. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], C. Kassel, [Quantum groups. Graduate Texts in Mathematics 155. New York: Springer-Verlag (1995; Zbl 0808.17003)], S. Majid, [Foundations of quantum group theory. Cambridge: Cambridge Univ. Press (1995; Zbl 0857.17009)], C. Kassel, M. Rosso and V. Turaev, [Quantum groups and knot invariants. Panoramas et Synthèses 5. Paris: Société Mathématique de France (1997; Zbl 0878.17013)], P. Etingof and O. Schiffmann, [Lectures on quantum groups. Lectures in Mathematical Physics. Boston: International Press (1998; Zbl 1105.17300)] and M. Aguiar and S. Mahajan, [Monoidal functors, species and Hopf algebras. CRM Monograph Series 29. Providence: AMS (2010; Zbl 1209.18002)].

Other areas not presented in this book are homology and cohomology of the braid groups, automatic structures on the braid groups, and applications to cryptography. This book grew out of the lectures given by the authors at the Bourbaki seminar in 1999 and 2000 and from graduate courses given by the first author at Université Louis Pasteur, Strasbourg, in 2002-2003 and by the second author at Indiana University, Bloomington, in 2006.

This book is a comprehensive introduction to the theory of braid groups. Assuming only a basic knowledge of topology and algebra, it is intended mainly for graduate and postdoctoral students. The book consists of seven chapters together with four short appendices on \(\mathrm{PSL}_2(\mathbb{Z})\), fibrations and homotopy sequences, Birman-Murakami-Wenzl algebras, and left self-distributive sets in order. Chapter 1 is concerned with the foundations of the theory of braids and braid groups. The central result of Chapter 2 is the Alexander-Markov description of oriented links in terms of Markov equivalence classes of braids. Chapter 3 is devoted to the Burau representation [introduced in W. Burau, Abh. Math. Semin. Hamb. Univ. 11, 179-186 (1935; Zbl 0011.17801)] and the Lawrence-Krammer-Bigelow representation [introduced in R. J. Lawrence, Commun. Math. Phys. 135, No. 1, 141-191 (1990; Zbl 0716.20022)]. Chapter 4 is concerned with the symmetry groups and the Iwahori-Hecke algebras. Chapter 5 deals with a classification of the finite-dimensional representations of the generic Iwahori-Hecke algebras in terms of Young diagrams. Chapter 6 is devoted to the Garside solution of the conjugacy problem in the braid groups. Chapter 7 presents the orderability of the braid groups, proving that the braid group \(B_n\) is orderable for every \(n\).

It is impossible to cover all the facets of the theory of braid groups in such a book of reasonable volume as this book. One important area entirely untouched is the connections with mathematical physics, quantum groups, Hopf algebras and braided monoidal categories, for which the reader is referred to G. Lusztig, [Introduction to quantum groups. Boston: Birkhäuser (1993; Zbl 0788.17010)], V. Chari and A. Pressley, [A guide to quantum groups. Cambridge: Cambridge University Press (1995; Zbl 0839.17010)], V. G. Turaev, [Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics 18. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], C. Kassel, [Quantum groups. Graduate Texts in Mathematics 155. New York: Springer-Verlag (1995; Zbl 0808.17003)], S. Majid, [Foundations of quantum group theory. Cambridge: Cambridge Univ. Press (1995; Zbl 0857.17009)], C. Kassel, M. Rosso and V. Turaev, [Quantum groups and knot invariants. Panoramas et Synthèses 5. Paris: Société Mathématique de France (1997; Zbl 0878.17013)], P. Etingof and O. Schiffmann, [Lectures on quantum groups. Lectures in Mathematical Physics. Boston: International Press (1998; Zbl 1105.17300)] and M. Aguiar and S. Mahajan, [Monoidal functors, species and Hopf algebras. CRM Monograph Series 29. Providence: AMS (2010; Zbl 1209.18002)].

Other areas not presented in this book are homology and cohomology of the braid groups, automatic structures on the braid groups, and applications to cryptography. This book grew out of the lectures given by the authors at the Bourbaki seminar in 1999 and 2000 and from graduate courses given by the first author at Université Louis Pasteur, Strasbourg, in 2002-2003 and by the second author at Indiana University, Bloomington, in 2006.

Reviewer: Hirokazu Nishimura (Tsukuba)

##### MSC:

20F36 | Braid groups; Artin groups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |

20C15 | Ordinary representations and characters |

20C08 | Hecke algebras and their representations |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

20F60 | Ordered groups (group-theoretic aspects) |

20M05 | Free semigroups, generators and relations, word problems |

55R80 | Discriminantal varieties and configuration spaces in algebraic topology |