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**Subfactors and knots. Expository lectures from the CBMS regional conference, held at the US Naval Academy, Annapolis, USA, June 5-11, 1988.**
*(English)*
Zbl 0743.46058

Regional Conference Series in Mathematics. 80. Providence, RI: American Mathematical Society (AMS). viii, 113 p. (1991).

The 9 lectures held at the CBMS Regional conference at Anapolis in 1988 give a beautiful introduction to the applications and ramifications of the structure theory for subfactors of finite index in a von Neumann factor, which has been initiated by the author (and for which he has received the Fields medal). They treat von Neumann algebras and knot theory as well as some elementary material from statistical mechanics and conformal field theory. Since the lectures were addressed to a mixed audience the author includes an introduction, in his refreshing and very clear style, to the basics of each of these subjects, making these notes very readable also for non-specialists.

The first lecture covers von Neumann algebras in general. The next three lectures contain a study of subfactors of finite index using Bratteli diagrams, and, in parallel, of representations of the Virasoro algebra developing certain analogies. In the next three lectures one learns about the braid group and its representations, knots and links, the new knot polynomial discovered by the author and other polynomials introduced subsequently. The last two lectures treat connections between knot theory and statistical mechanics and, finally, Hecke algebras and the Birman- Murakami-Wenzl algebra and their connections with knot invariants.

These notes are a must for anybody who wants to familiarize himself with this fascinating circle of ideas. But also for the expert they give new insights and present many ideas more clearly than the original articles.

The first lecture covers von Neumann algebras in general. The next three lectures contain a study of subfactors of finite index using Bratteli diagrams, and, in parallel, of representations of the Virasoro algebra developing certain analogies. In the next three lectures one learns about the braid group and its representations, knots and links, the new knot polynomial discovered by the author and other polynomials introduced subsequently. The last two lectures treat connections between knot theory and statistical mechanics and, finally, Hecke algebras and the Birman- Murakami-Wenzl algebra and their connections with knot invariants.

These notes are a must for anybody who wants to familiarize himself with this fascinating circle of ideas. But also for the expert they give new insights and present many ideas more clearly than the original articles.

Reviewer: J.Cuntz (Heidelberg)

### MSC:

46L10 | General theory of von Neumann algebras |

46L35 | Classifications of \(C^*\)-algebras |

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

46L37 | Subfactors and their classification |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

57M15 | Relations of low-dimensional topology with graph theory |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

46L60 | Applications of selfadjoint operator algebras to physics |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

82B23 | Exactly solvable models; Bethe ansatz |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |