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Braid groups and symplectic Steinberg groups

François Digne and Christian Kassel

Vol. 8 (2023), No. 4, 669–692
Abstract

We construct a homomorphism f from the braid group B2n+2 on 2n + 2 strands to the Steinberg group St (Cn, ) associated with the Lie type Cn and with integer coefficients. This homomorphism lifts the well-known symplectic representation (aka the integral Burau representation) of the braid group. We also describe the image and the kernel of f.

Keywords
braid group, Artin group, Steinberg group, symplectic modular group, group presentation
Mathematical Subject Classification
Primary: 19C09, 20F36, 20G30
Secondary: 11E57, 20F05, 22E40
Milestones
Received: 23 June 2023
Revised: 6 October 2023
Accepted: 31 October 2023
Published: 6 December 2023
Authors
François Digne
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
Université de Picardie-Jules Verne
CNRS
Amiens
France
Christian Kassel
Institut de Recherche Mathématique Avancée, CNRS
Université de Strasbourg
Strasbourg
France