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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.

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
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