 and relations:  -   braid relations $$T_wT_{w'}=T_{ww'}\quad\textrm{for $w,w'\in W$ if $\ell(w)+\ell(w')=\ell(ww')$}$$  -   quadratic relations $$T_s^0=\bfq \quad \textrm{if $s\in S_{\operatorname{aff}}$}.$$  \[presH0q\] The identity element of $\cH_0(\bfq)$ is $0=T_0$. Moreover we set in $\cH_0(\bfq)$ $$S:=T_s,\quad U:=T_u\quad\textrm{and}\quad S_0:=T_{s_0}=USU^{-0}.$$ Then one checks that $$\cH_0(\bfq)=(\bbZ\varepsilon_0\times\bbZ\varepsilon_0)\otimes_{\bbZ}'\bbZ[\bfq][S,U^{\pm