In fact this has been clearly stated in the follow paragraph of <PERSON>'s note.
Since two independent algebras naturally arise, it is frequently useful to regard z and z as independent coordinates. (More formally, we would say that since the action of the conformal group in two dimensions factorizes into independent actions on z and z, Green functions of a 2d conformal field theory may be continued to a larger domain in which z and z are treated as independent variables.) In terms of the original coordinates (x1, x2) ∈ $R^2$, this amounts to taking instead (x1, x2) ∈ C2, and then the transformation to z, z coordinates is just a change of variables. In C2, the surface defined by z = z∗ is the ‘real’ surface on which we recover (x, y) ∈ R2. This procedure allows the algebra A ⊕ $\overline{A}$ to act naturally on C2, and the ‘physical’ condition z = z∗ is left to be imposed at our convenience. The real surface specified by this condition is preserved by the subalgebra of A⊕$\overline{A}$ generated by ℓn+ℓn and i(ℓn−ℓn). 469
In the sections that follow, we shall frequently use the independence of the algebras A and A to justify ignoring anti-holomorphic dependence for simplicity, then reconstruct it afterwards by adding terms with bars where appropriate.
(sorry that I do not type tex command but just copy the above context from pdf)
for example consider the generator for rotation operation, denoted as $l$ . Points in the total space is $(z,\overline{z})$, acting $\exp(i\epsilon l)$ on it just let it goes to $(e^{i\epsilon}z,\overline{z})$ which only rotating $z$ but do not rotate $\overline{z}$. To actually rotate $(x,y)$ you should in fact act $l+\overline{l}$ which both changing $z$ and $\overline{z}$. However, since it's obvious that the change of $\overline{z}$ could be related to those of $z$ when we restrict the total space on the "physical" plane, so we usually only talk about how $z$ changes.
In a word, $\overline{l}$ has nothing to do with the anti-holomorphic transformation. It's how $\overline{z}$ changes under holomorphic transformation. 976