Braid groups and symplectic Steinberg groups

TL;DR

This paper constructs a homomorphism from braid groups to symplectic Steinberg groups, extending the symplectic representation and analyzing its algebraic properties.

Contribution

It introduces a new homomorphism linking braid groups to symplectic Steinberg groups and characterizes its image and kernel.

Findings

Homomorphism from braid groups to symplectic Steinberg groups constructed

Image and kernel of the homomorphism explicitly described

Lifts the classical symplectic representation of braid groups

Abstract

We construct a homomorphism $f$ from the braid group $B_{2n+2}$ on $2n+2$ strands to the Steinberg group associated with the Lie type $C_n$ and with integer coefficients. This homomorphism lifts the well-known symplectic representation of the braid groups. We also describe the image and the kernel of $f$.