Hall algebras and quantum groups associated to Dynkin quivers

For Dynkin quivers, we find the Laurent polynomials $\widetilde{X}{a, c}^{b}(v)$anduse$\widetilde{X}{a, c}^{b}(v)$toconstructtheHallalgebra$\hc_v(\cc(\cp))$over$\mz[v, v^{-1}]$,where$\widetilde{X}_{a, c}^{b}(|\mf_q|)$'s are structure constants used by Bridgeland. The Laurent polynomials ${\tilde{X}}_{a,c}^b(v)$ are explicitly given in $A_1$ case. As an application, we obtain the full quantum groups $U_t(\text{sg})$ associated to the Dynkin quivers for arbitrary $t≠0,\pm 1$.