nLab Drinfeld associator

Definition

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

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

   in $\widehat{U(L_4)}$

2. the hexagon equation
   $$\exp(\lambda a/2)\Phi(c,a)\exp(\lambda c/2)\Phi(b,c)\exp(\lambda b/2)\Phi(a,b)=1$$

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

Remark

The set of “0-associators” is 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.

Theorem

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

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

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

Remark

Examples of metrizable Lie algebras are provided by simple Lie algebras, in which case $t$ 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.

Existence

An explicit associator over $\mathbf{C}$ 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 $\mathbf{Q}$ also exists.

Applications

• Drinfeld-Kohno theorem
• Etingof-Kazhdan quantization functor?
• Formality? of the little 2-disk operad
• Construction of a functorial universal finite type invariant?
• dilogarithm
• Hausdorff series, Kashiwara-Vergne conjecture

References

• Vladimir Drinfeld. On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal($\overline{\mathbf{Q}}/\mathbf{Q}$). Leningrad Math. J. 2:4 (1990), 829–860. (mathnet.ru)
• Dror Bar-Natan. On associators and the Grothendieck-Teichmuller group I. Selecta Math. (N.S.) 4:2 (1998), 183–212. (pdf)
• A. Alekseev, C. Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators, Ann. of Math. 175 (2012), no.2, 415–463 pdf; Kontsevich deformation quantization and flat connections arxiv/0906.0187 doi
• A. Alekseev, B. Enriques, C. Torossian, Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations, Publ. Math. Inst. Hautes ́Etudes Sci. 112 (2010), 143–189 arxiv/0903.4067
• Anton Alekseev, Florian Naef, Xiaomeng Xu, Chenchang Zhu, Chern–Simons, Wess–Zumino and other cocycles from Kashiwara–Vergne and associators, Letters in Mathematical Physics 108:3 (2018) 757–778 doi