  The question of whether a representation of Artin's pure braid group is
faithful is translated to certain properties of the Lie algebra arising from
the descending central series of the pure braid group, and thus the Vassiliev
invariants of pure braids via work of T. Kohno \cite{kohno1,kohno2}. The main
result is a Lie algebraic condition which guarantees that a homomorphism out of
the classical pure braid group is faithful. However, it is unclear whether the
methods here can be applied to any open cases such as the Gassner
representation.
