Sheaf realization of Bridgeland's Hall algebra of Dynkin type

As one of results in [6], Bridgeland realized the quantum group ${\mathbf{U}}_v$ via the localization of Ringel-Hall algebra for the two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig’s categorical construction for the nilpotent part ${\mathbf{U}}_v^{+}$ to Bridgeland’s Hall algebra of Dynkin type. In particular, we obtain a basis of the Ringel-Hall algebra for the two-periodic projective complexes which has the positivity, and we categorify an integral form of the generic Bridgeland’s Hall algebra which is isomorphic to the Poisson integral form of ${\mathbf{U}}_v$, and obtain a $\mathbb{Z}[v,v^{-1}]$-basis of this integral form.