Drinfeld associator



Let k\mathbf{k} be a field of characteristic 0 and λk *\lambda \in \mathbf{k}^*. A λ\lambda-Drinfeld associator, or just λ\lambda-associator, is a grouplike element Φ(a,b)\Phi(a,b) of the k\mathbf{k}-algebra of formal power series in two non-commuting variables a,ba,b satisfying:

  1. The pentagon equation
    Φ(t 12,t 23+t 24)Φ(t 13+t 23,t 34)=Φ(t 23,t 34)Φ(t 12+t 13,t 24+t 34)Φ(t 12,t 23) \Phi(t_{12} , t_{23} + t_{24} )\Phi(t_{13} + t_{23} , t_{34} ) = \Phi(t_{23} , t_{34} )\Phi(t_{12} + t_{13} , t_{24} + t_{34} )\Phi(t_{12} , t_{23} )

    in U(L 4)^\widehat{U(L_4)}

  2. the hexagon equation
    exp(λa/2)Φ(c,a)exp(λc/2)Φ(b,c)exp(λb/2)Φ(a,b)=1 \exp(\lambda a/2)\Phi(c,a)\exp(\lambda c/2)\Phi(b,c)\exp(\lambda b/2)\Phi(a,b)=1

where L 4L_4 is the fourth Drinfeld-Kohno Lie algebra and c=abc=-a-b.


The set of “0-associators” is the what is called the Grothendieck-Teichmueller group. This acts freely on the set of Drinfeld associators.

Relations with braided monoidal categories

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 (𝔤,t)(\mathfrak{g},t) be a metrizable Lie algebra, that is a Lie algebra 𝔤\mathfrak{g} together with a non-degenerate symmetric 𝔤\mathfrak{g}-invariant 2-tensor tt. Then if Φ\Phi is a λ\lambda-associator and \hbar a formal variable, then the action of

Φ((t1),(1t))U(𝔤) 3[[]] \Phi(\hbar (t \otimes 1),\hbar (1\otimes t)) \in U(\mathfrak{g})^{\otimes 3}[[\hbar]]

and e λt/2Pe^{\hbar \lambda t/2}\circ P turns the category of U(𝔤)[[]]U(\mathfrak{g} ) [ [ \hbar ] ] module into a braided monoidal category, where PP is the flip: P(ab)=baP(a\otimes b)=b\otimes a.


Examples of metrizable Lie algebras are provided by simple Lie algebras, in which case tt 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 C\mathbf{C} 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 Q\mathbf{Q} also exists.



  • Vladimir Drinfeld. On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q¯/Q\overline{\mathbf{Q}}/\mathbf{Q}). Leningrad Math. J. 2:4 (1990), 829–860. (

  • Dror Bar-Natan. On associators and the Grothendieck-Teichmuller group I. Selecta Math. (N.S.) 4:2 (1998), 183–212. (pdf)

Last revised on April 17, 2018 at 13:37:15. See the history of this page for a list of all contributions to it.