The Drinfeld associator $\Phi(x\_0, x\_1)$ encodes the parallel transport of the Knizhnik-Zamolodchikov (KZ) connection $\nabla$ on the bundle $\mathbb{C}\langle\langle x\_0, x\_1\rangle\rangle$ of formal power series in noncommutating variables $x\_0, x\_1$
over $X:=\mathbb{P}^1(\mathbb{C})\backslash\left\{0,1,\infty\right\}$. Writing
$$\omega\_0=\frac{dz}{z}, \quad \omega\_1 = \frac{dz}{1-z},$$
we can express the KZ connection as 
$$\nabla = d-\omega\_{KZ}$$
where $\omega\_{KZ}\in\Omega^1(X)\otimes\mathbb{C}\langle x\_0, x\_1\rangle$ is the ''connection form''
$$\omega\_{KZ} = x\_0\omega\_0 + x\_1\omega\_1.$$


The Drinfeld associator $\Phi(x\_0,x\_1)\in\mathbb{C}\langle\langle x\_0, x\_1\rangle\rangle$ (sometimes called KZ associator) is the result of analytically continuing a global horizontal section of $\nabla$ taking the asymptotic value $1$ at $z=0$ along the ''straight line path'' $\text{dch}(t)=t$, $0 < t < 1$. In other words, it is the parallel transport along $\text{dch}$. It can be expressed using iterated integrals as
$$\Phi(x\_0,x\_1) = 1 + \int\_{\text{dch}}\omega\_{KZ} + \int\_{\text{dch}}\omega\_{KZ}\omega\_{KZ}+\dots$$


It is well-known (e.g. see Lemma 4.2 of Brown's *[Iterated integrals in quantum field theory](http://www.ihes.fr/~brown/ColombiaNotes7.pdf)*) that the coefficients of the Drinfeld associator are multiple zeta values (MZVs). Specifically, we can write
$$\Phi(x\_0,x\_1) = \sum\_{w \text{ word in } x\_0, x\_1}\zeta(w)w$$
where $\zeta(w)$ is the ''shuffle-regularised'' MZV such that
$$\zeta(x\_0^{k\_1-1}x\_1\dots x\_0^{k\_n-1}x\_1) = \zeta(k\_1, \dots,k\_n), \quad \zeta(x\_0) = \zeta(x\_1)=0.$$
(This depends on your conventions for writing iterated integrals etc.)


The inverse of the Drinfeld associator (as power series in noncommuting variables) is
$$\Phi(x\_1,x\_0) = \Phi(x\_0,x\_1)^{-1}.$$
I take this to mean
$$\Phi(x\_1,x\_0) = \sum\_{w \text{ word in } x\_0, x\_1}\zeta(\overline{w})w$$
where $\overline{w}$ denotes $w$ with $x\_0$ and $x\_1$ swapped, and the zeta value requires shuffle-regularising.



> 
> Is there a simple formula for $\Phi(x\_1,x\_0) = \Phi(x\_0,x\_1)^{-1}$ in terms of MZVs without having to apply a combinatorial action like swapping to the words? The desire to not deal with things like $\zeta(\overline{w})$ is that the shuffle-regularistion procedure is very combinatorially complicated. 
> 
> 
> 


As an example of what I mean by ''simple formula'', I originally thought perhaps
$$\Phi(x\_1,x\_0)=\sum\_{w} (-1)^{\text{weight}(w)}\zeta(w)w, \quad \text{weight}(w) := \text{length of } w.$$
However this is not correct, because we know that $\zeta(2)=\zeta(x\_0 x\_1) = -\zeta(x\_1 x\_0)$ (using shuffle-regularisation). I expect that some formula would use the path-reversal formula for iterated integrals (because we are now integrating along $\text{dch}^{-1}$) and perhaps the change of variables $z\mapsto 1-z$ on $X$.



Is there a ''simple'' formula for the inverse of the Drinfeld associator?
Try


\begin{equation}
\Phi(x\_1,x\_2)=\sum\_w \left(-1\right)^{\textrm{weight}(w)} \zeta(w) \tilde{w} 
\end{equation}


where $\tilde{w}$ is the word with reversed order. Because the $\zeta(w)$ obey the shuffle product, that should give you a proper inverse. Try the first few orders! 


