Geodesic
Parametrized braid groups of Chevalley groups.
Documenta mathematica, Tome 10 (2005), pp. 391-416
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We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct product of the Steinberg group (of the ring) with the classical braid group. The technical heart of the proof is the Pure Braid Lemma, which asserts that certain elements of the parametrized braid group commute with the pure braid group. This first part treats the case of the root system An​; in the second part we prove a similar theorem for the root system Dn​.
DOI : 10.4171/dm/195
Classification : 19Cxx, 20F36, 20F55
Mots-clés : braid group, root system, Steinberg group, parametrized braid group 
@article{10_4171_dm_195,
     author = {Jean-Louis Loday and Michael R. Stein},
     title = {Parametrized braid groups of {Chevalley} groups.},
     journal = {Documenta mathematica},
     pages = {391--416},
     year = {2005},
     volume = {10},
     doi = {10.4171/dm/195},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/195/}
}
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Jean-Louis Loday; Michael R. Stein. Parametrized braid groups of Chevalley groups.. Documenta mathematica, Tome 10 (2005), pp. 391-416. doi: 10.4171/dm/195

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