  We study a quotient of the group algebra of the braid group in which the
Artin generators satisfy a cubic relation. This quotient is maximal among the
ones satisfying such a cubic relation. It is finite-dimensional for at least n
at most 5 and we investigate its module structure in this range. We also
investigate the proper quotients of it that appear in the realm of quantum
groups, and describe another maximal quotient related to the usual Hecke
algebras. Finally, we describe the connection between this algebra and a
quotient of the algebra of horizontal chord diagrams introduced by Vogel. We
prove that these two are isomorphic for n at most 5.
