symmetric monoidal (∞,1)-category of spectra
where is the fourth Drinfeld-Kohno Lie algebra and .
The set of “0-associators” is the what is called the Grothendieck-Teichmueller group. This acts freely on the set of Drinfeld associators.
These equations are modelled on the defining axioms of braided monoidal categories. Indeed, associators provides a universal way of constructing braided monoidal categories out of some Lie algebraic data.
Drinfeld associators are also used to construct quasi-Hopf algebras.
Let be a metrizable Lie algebra, that is a Lie algebra together with a non-degenerate symmetric -invariant 2-tensor . Then if is a -associator and a formal variable, then the action of
and turns the category of module into a braided monoidal category, where is the flip: .
Examples of metrizable Lie algebras are provided by simple Lie algebras, in which case is a scalar mutliple of the Killing form. The braided monoidal category obtained this way is equivalent to that constructed from the corresponding quantum group, by the Drinfeld-Kohno theorem.
An explicit associator over was constructed by Drinfeld from the monodromy of a universal version of the Knizhnik-Zamolodchikov equation. Using the non-emptiness of the set of associators, and the fact that is is a torsor under the action of the Grothendieck-Teichmueller group, he show that associators over also exists.