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.



What kind of object are the solutions of the Knizhnik-Zamolodchikov Equations
There are really 3 levels at which you can define this equation:


1. 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 $V^{\otimes n}[[h]]$.
2. 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 $h$).
3. 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 $\tilde G(z\_0)=v$.


