On the representation theory of braid groups
[Sur les représentations des groupes de tresses]
Annales Mathématiques Blaise Pascal, Tome 20 (2013) no. 2, pp. 193-260.

Ce travail présente une approche en direction d’une théorie des représentations des groupes de tresses B n . Nous nous concentrons sur les représentations de dimensions finie sur le corps des séries de Laurent qui peuvent être obtenues à partir des représentations des tresses infinitésimales en utilisant des associateurs de Drinfeld. Nous établissons un dictionnaire entre les propriétés de théorie des représentations de ces deux structures, ainsi que des outils pour décrire les représentations ainsi obtenues. Nous expliquons l’apparition fréquente de structures unitaires préservées par les représentations classiques. Nous introduisons de nouveaux objets (variétés d’extensions tressées, quotients infinitésimaux) qui sont utiles dans ce cadre, et nous analysons plusieurs de leurs propriétés. Enfin, nous passons en revue les représentations les plus classiques des groupes de tresses, montrons comment elles peuvent être obtenues par nos méthodes et comment ce cadre enrichit la compréhension que l’on en a.

This work presents an approach towards the representation theory of the braid groups B n . We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids, with the help of Drinfeld associators. We set a dictionary between representation-theoretic properties of these two structures, and tools to describe the representations thus obtained. We give an explanation for the frequent apparition of unitary structures on classical representations. We introduce new objects — varieties of braided extensions, infinitesimal quotients — which are useful in this setting, and analyse several of their properties. Finally, we review the most classical representations of the braid groups, show how they can be obtained by our methods and how this setting enriches our understanding of them.

DOI : https://doi.org/10.5802/ambp.326
Classification : 20C99,  20F36
Mots clés: Linear representations, Braid groups
@article{AMBP_2013__20_2_193_0,
     author = {Ivan Marin},
     title = {On the representation theory of braid groups},
     journal = {Annales Math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {20},
     number = {2},
     year = {2013},
     pages = {193-260},
     doi = {10.5802/ambp.326},
     zbl = {06251800},
     mrnumber = {3138029},
     language = {en},
     url = {ambp.centre-mersenne.org/item/AMBP_2013__20_2_193_0/}
}
Ivan Marin. On the representation theory of braid groups. Annales Mathématiques Blaise Pascal, Tome 20 (2013) no. 2, pp. 193-260. doi : 10.5802/ambp.326. https://ambp.centre-mersenne.org/item/AMBP_2013__20_2_193_0/

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