What kind of object are the solutions of the Knizhnik-Zamolodchikov Equations I am reading about the KZ equations in Kassel's Quantum groups. In definition XIX.3.1 (page 455) he defines the differential system $(KZ_n)$ as 
$$
dw = \frac{h}{2\pi\sqrt{-1}} \sum_{1 \leq i <j \leq n} \frac{t_{ij}}{z_i-z_j}(dz_i- dz_j)w $$
where the  $t_{ij}$ are some element of the universal enveloping algebra $U(\mathfrak{g})$ for some Lie algebras $\mathfrak{g}$. In this definition "$w = w(z_1, \dots, z_n)$ is a function on $Y_n$ with values in $W^{\otimes n}$" where $Y_n = \{(z_1, \dots, z_n) \vert  i \neq j \Rightarrow z_i \neq z_j\}$ and $V$ is a finite dimensional $\mathfrak{g}$-module.
At this point the solutions are then simply foncitons on some complex parameters taking value in the $V^{\otimes n}$. However, when doing the proof of the Drinfled-Kohno theorem , Kassel makes some change of coordinates for the case $n=3$ and we have the following equation (equation 7.3, page 469 in Kassel's book): 
$$ w(z_1,z_2,z_3) = (z_3 - z_1)^{\hbar(t_{12} + t_{23} + t_{13})}G(z)$$
where $z = \frac{z_2-z_1}{z_3-z_1}$ and $G(z)$ is a formal serie in $t_{12}$ and $t_{23}$ with coefficients being analytic functions in $z$. This seems to indicate that the solutions are actually taking value in in $U(\mathfrak{g})^{\otimes 3} [[h]]$ and this is where my confusion comes from. Especially since it does not seem to be coming from the case where $V = U(\mathfrak{g})$ for this is not finite dimensional.
Maybe the explanation for this is in the paragraph at the end of page 457 but I don't really understand what he is doing here neither.
 A: There are really 3 levels at which you can define this equation:


*

*for functions
$$G:Y_n \longrightarrow V^{\otimes n}$$ where $V$ is a f.d. module and $h$ is a complex number. You can then take a formal expansion around $h=0$ and regard $G$ as being valued in $V^{\otimes n}[[h]]$.

*for functions
$$G:Y_n \longrightarrow U(\mathfrak g)^{\otimes n}[[h]].$$
This only makes sense formally (by which I mean that you need formal power series in $h$). 

*for $n=3$ and up to some change of variable as you say, you get a version where $G$ takes values in the algebra of formal power series in two non commuting variables $A,B$. This is the one appearing in section 6 of Kassel's book. More generally, for arbitrary $n$ you have a version of the KZ equation taking values in the algebra of so-called horizontal chords diagrams.


The general theory of this kind of equation really has been done only in case 1, so in the other 2 cases, in the literature, existence and uniqueness of solutions are proved in a somewhat ad hoc way but this isn't hard. 
The relation between 3 and 2 just boils down to the substitution $A\mapsto h t_{1,2}, B\mapsto h t_{2,3}$. So in equation 7.3 that you mention in your message, $G$ does indeed takes value in $U(\mathfrak g)^{\otimes 3}[[h]]$ (so it's a formal power series in $h$, not in $t_{1,2}$ and $t_{2,3}$).
The relation between 2 and 1 at the level of equations is that the former specializes to the latter through the algebra map
$$U(\mathfrak g)^{\otimes n}[[h]] \rightarrow End(V^{\otimes n}[[h]])$$
coming from the action of $\mathfrak g$ on $V$.
In terms of solutions, if $z_0 \in Y_n$ then there is a unique solution $G$ of $2$ defined in a neighborhood of $z_0$ such that $G(z_0)=1$, the unit of $U(\mathfrak g)$. Now if $v \in V^{\otimes n}$, then the function $$\tilde G:z\longmapsto G(z)\cdot v$$
is by construction the unique solution of equation 1 satisfying $\tilde G(z_0)=v$.
