# nLab Drinfeld associator

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 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.

###### 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.

## 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