Braid groups and symplectic Steinberg 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 $f$.