The generalized Cartan matrix module-algebra conjecture for Hecke-Hopf algebras

Let $A=(a_{ij})$, $i,j\in I$, be a generalized Cartan matrix, and let $W=W_A$ be the corresponding crystallographic Coxeter group, with

$$m_{ij}=\begin{cases}2+a_{ij}a_{ji} & \text{if }a_{ij}a_{ji}\le 2,\\6 & \text{if }a_{ij}a_{ji}=3,\\0 & \text{if }a_{ij}a_{ji}>3,\end{cases}$$

for $i\ne j$. Let ${\mathcal L}_I=\mathbb{Z}[t_i^{\pm1}\mid i\in I]$. The generalized Cartan matrix module-algebra conjecture. The assignments

$$s_i(t_j)=t_i^{-a_{ij}}t_j,\qquad D_i(t_j)=t_j\frac{1-t_i^{-a_{ij}}}{1-t_i}$$

for $i,j\in I$ turn ${\mathcal L}_I$ into an ${\bf H}(W)$-module algebra. This is presented as a conjectural generalization of the corresponding result for the previously treated cases; its validity for arbitrary generalized Cartan matrices is not established in the supplied text.