On the representation theory of braid groups

Ivan Marin[1]

  • [1] LAMFA Université de Picardie-Jules Verne 33 rue Saint-Leu 80039 Amiens Cedex 1 France

Annales mathématiques Blaise Pascal (2013)

  • Volume: 20, Issue: 2, page 193-260
  • ISSN: 1259-1734

Abstract

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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.

How to cite

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Marin, Ivan. "On the representation theory of braid groups." Annales mathématiques Blaise Pascal 20.2 (2013): 193-260. <http://eudml.org/doc/275607>.

@article{Marin2013,
abstract = {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.},
affiliation = {LAMFA Université de Picardie-Jules Verne 33 rue Saint-Leu 80039 Amiens Cedex 1 France},
author = {Marin, Ivan},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Linear representations; Braid groups; representations; braid groups; classical braid representations; Drinfeld associators; infinitesimal braids; completions; power series},
language = {eng},
month = {7},
number = {2},
pages = {193-260},
publisher = {Annales mathématiques Blaise Pascal},
title = {On the representation theory of braid groups},
url = {http://eudml.org/doc/275607},
volume = {20},
year = {2013},
}

TY - JOUR
AU - Marin, Ivan
TI - On the representation theory of braid groups
JO - Annales mathématiques Blaise Pascal
DA - 2013/7//
PB - Annales mathématiques Blaise Pascal
VL - 20
IS - 2
SP - 193
EP - 260
AB - 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.
LA - eng
KW - Linear representations; Braid groups; representations; braid groups; classical braid representations; Drinfeld associators; infinitesimal braids; completions; power series
UR - http://eudml.org/doc/275607
ER -

References

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