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Développements Récents sur les Groupes de Tresses Applications à la Topologie et à l’Algèbre

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Functional Integration

Part of the book series: NATO ASI Series ((NSSB,volume 361))

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Abstract

Braid groups were invented by Emil Artin in 1925. The most convenient definition of the braid group B n (n is the number of strings) is as the fundamental group of the configuration space Φ n parametrizing the subsets with n points in a given pane. From a physical point of view, the space Φ n is the configuration space of n indistinguishable particles moving in two dimensions. This explains the recent interest for the braid groups in many two-dimensional physical models, in the form of parastatistics and anyons.

From a more mathematical point of view, Markov discovered in 1936 how to construct invariants of knots from linear representations of the braid groups. For a long time, very few representations of the braid groups were known. One of the greatest recent discoveries has been the invention of the Jones polynomial for knots (in 1984). The corresponding representations of B n were introduced in connection with the von Neumann algebras in operator theory, or in another form, via solutions of the Yang—Baxter equation.

In this paper, after a recollection of known facts about the combinatorics of knots, we develop consequently a categorical point of view. For a long time, categories — in the form described in the 40’s by Eilenberg and MacLane — were collections of preexisting mathematical objects connected via certain kind of transformations. In the newest developments, categories appear to be more abstract, defined via generators and relations, and encapsulating a great deal of sophisticated combinatorics.

We report on the main results of Kohno, Drinfeld and Reshetikhin, which provide a link between the main two approaches to knot invariants:

  • through the monodromy of the so-called Knizhnik—Zamolodchikov differential equations

  • or using representations of quantum groups.

After the writing of this paper—in 1989—the most dramatic developments were the definition of the Vasiliev invariants, and their explicit description by Kontsevitch, Bar-Natan and myself in 1993.

A mes amis soviétiques retrouvés grâce à la “perestroïka”

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  1. Département de Mathématique et Informatique, École Normale Supérieure, 45 rue d’Ulm, F-75230, Paris Cedex 05, France

    P. Cartier

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  1. The University of Texas at Austin, Austin, Texas, USA

    Cecile DeWitt-Morette

  2. Ecole Normale Supérieure, Paris, France

    Pierre Cartier

  3. Université de Corse, Corte, France

    Antoine Folacci

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Cartier, P. (1997). Développements Récents sur les Groupes de Tresses Applications à la Topologie et à l’Algèbre. In: DeWitt-Morette, C., Cartier, P., Folacci, A. (eds) Functional Integration. NATO ASI Series, vol 361. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0319-8_8

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