Let . The Drinfel’d-Kohno Lie algebra is a -algebra defined by generators , subject to the relations
It is the holonomy Lie algebra? of the configuration space of distincts points in the complex plane. Hence, it can be used to define a flat connection on , which is universal among Knizhnik-Zamolodchikov equations.
Therefore, it induces a monodromy representation of which is isomorphic to the pure braid group:
It was shown by Kohno that the extension of this map to the -pro-unipotent completion of is an isomorphism. Drinfeld showed using associators that the same holds true over .
In particular, is isomorphic to the associated graded of with respect to the filtration induced by powers of the augmentation ideal. Since it is known that this filtration coincides with the one induced by the Vassiliev skein relation, may be identified with the algebra of horizontal chord diagrams.
The universal enveloping is a Koszul algebra.