I don't fully understand the definition of root vectors in the construction of the so called Andruskiewitsch-Schneider Hopf-algebras. See for example section 2 of the paper 'On the Classification of Finite-Dimensional Pointed Hopf-algebras' by Andruskiewitsch and Schneider. The definition should be similar to the definition of root vectors for Drinfeld-Jimbo algebras (or even root vectors for ordinary universal enveloping algebras). I will briefly recall these definitions in the first part of the question as a reminder of what it should be. In the second part I will briefly discuss the definition of the Andruskiewitsch-Schneider Hopf-algebras up to the point where I am lost.
Root vectors of Drinfeld-Jimbo algebras
It is well known that if $\mathfrak{g}$ is a (complex) semisimple Lie algebra and $\alpha_1, \dots, \alpha_l$ are simple roots, then the Weyl group $W$ is generated by the corresponding reflections $w_{\alpha_i}\equiv w_i$. Thus any $w\in W$ can be decomposed as a product of the reflections $w_i$. A decomposition $w=w_{i_1}\dots w_{i_k}$ is called reduced if $k$ is minimal. The following proposition is well known.
Let $w_0$ be an element of maximal length in $W$ and $w_0=w_{i_1}\dots w_{i_n}$ a reduced decomposition, then we define roots $\beta_1, \dots ,\beta_n$ by $$\beta_r=w_{i_1}\dots w_{i_{r-1}}(\alpha_{i_r}).$$ The roots $\beta_1, \dots ,\beta_n$ are positive and pairwise distinct, moreover, they exhaust all positive roots of $\mathfrak{g}$.
One can also consider the braid group $\mathfrak{B_g}$ which can be defined as the group generated by $s_1, \dots , s_l$ which satisfy the same relations as the reflections in the Weyl group, but we forget that they are reflections, i.e. we drop the relation $s_i^2=1$. It's well known that $\mathfrak{B_g}$ acts on the quantized universal enveloping algebra (Drinfeld-Jimbo algebra) $U_q(\mathfrak{g})$. In fact to each $i=1, 2, \dots ,l$, we have algebra automorphisms $\mathcal{T}_i$ which can be defined explicitly on the generators of $U_q(\mathfrak{g})$. (See for example theorem $22$ in 'Quantum groups and their representations' by Klimyk and Schmüdgen).
Using all of this, one can define the so called root vectors:
The elements $E_{\beta_r}:=\mathcal{T}_{i_1}\mathcal{T}_{i_2}\dots \mathcal{T}_{i_{r-1}}(E_{i_r})$ are called root vectors corresponding to the roots $\beta_r$. Using the same action on $F_{i_r}$ one obtains the root vector corresponding to $-\beta_r$.
(Remark: Notice that the above definition uses the decomposition of longest element of the Weyl group and not the braid group. In a similar fashion one can define root vectors for $U(\mathfrak{g})$ using only the Weyl group. One can show that different decompositions of $w_0$ give rise to the same root vectors. This is no longer true for $U_q(\mathfrak{g})$.)
The point of these root vectors is that we can get a nice analogue of the Poincaré-Birkhoff-Witt theorem. In particular we get vector space bases in terms of the $E_{\beta_r}^k,F_{\beta_{r'}}^l$ and $K_i^j$ with the $E$'s collected at one side and the $F$'s at the other side.
Andruskiewitsch-Schneider Hopf-algebras
First we need a datum of Cartan type, that is $$\mathcal{D}=(\Gamma,g_i,\chi_i,A)$$ where $\Gamma$ is an abelian group, $g_i\in \Gamma$, $\chi_i$ are characters and $A$ is a generalized Cartan matrix such that $q_{ij}:=\chi_j(g_i)$ and $$q_{ij}q_{ji}=q_{ii}^{a_{ij}}$$ for all $1\leq i,j\leq \theta$. One defines a Weyl group as the group generated by the reflections $s_i:\mathbb{Z}^{\theta}\rightarrow \mathbb{Z}^{\theta}$ defined by $s_i(\alpha_j):=\alpha_j-a_{ij}\alpha_i$. Here the $\alpha_i$ are the free generators of $\mathbb{Z}^{\theta}$. The root system $\Phi$ is defined by $$\Phi=\bigcup_{i=1}^{\theta}W(\alpha_i)$$ and the positive roots $\Phi^+$ are those roots which can be written as a positive integral combination of the $\alpha_i$'s.
Now one considers a Yetter-Drinfeld module $V(\mathcal{D})\in {^{k\Gamma}_{k\Gamma}\mathcal{YD}}$ defined by $$V(\mathcal{D})=\sum_{i=1}^{\theta}V_{g_i}^{\chi_i},$$ where $V_{g_i}^{\chi_i}=k\left\{X_i\right\}$ is the $1$-dimensional Yetter-Drinfeld module defined by $$\delta(X_i)=h_i\otimes X_i \mbox{ and } g\cdot X_i=\chi_i(g)X_i.$$ The tensor algebra $T(V(\mathcal{D}))$ is a braided Hopf-algebra in ${^{k\Gamma}_{k\Gamma}\mathcal{YD}}$ (the $X_i$'s are primitive elements and the braiding is determined by $X_i\otimes X_j\mapsto q_{ij}X_j\otimes X_i$). It's well known that $(\text{ad}_cX_i)^{1-a_{ij}}(X_j)$ are primitive elements and they generated a braided Hopf ideal $I$. Thus we get a quotient braided Hopf algebra $\mathcal{B(D)}=T(V(\mathcal{D}))/I$.
Again we can take a longest element $w_0$ in the Weyl group. In a similar fashion as before we obtain all positive roots as $\beta_r=s_{i_1}\dots s_{i_{r-1}}(\alpha_{i_r})$. Using this one defines for each $\beta\in \Phi^+$ a root vector $X_{\beta}\in\mathcal{B(D)}$.
This last line is where I am lost. Andruskiewitsch-Schneider always refer to the paper "Quantum groups at roots of $1$" by Lusztig for the proper definition of $X_{\beta}$. But looking at that paper I can only see a definition in the case of quantized universal enveloping algebras $U_q(\mathfrak{g})$ for which I understand the definition of root vectors.
Edit: I believe the following definition makes sense. Let $\beta_r$ be a positive root vector as before. Then we define $X_{\beta_r}$ as $$X_{\beta_r}=(\text{ad}_cX_{i_1})\circ (\text{ad}_cX_{i_2})\dots \circ (\text{ad}_cX_{i_{r-1}})(X_{i_r}).$$ Is this correct?