Sheaf realization of Bridgeland's Hall algebra of Dynkin type

Abstract

As one of results in [6], Bridgeland realized the quantum group $\mathrm{U}_v(\mathfrak{g})$ via the localization of Ringel-Hall algebra for two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig's categorical construction and (dual) canonical basis for the nilpotent part $\mathrm{U}_v(\mathfrak{n}^+)$ to Bridgeland's Hall algebra of Dynkin type, and obtain a perverse sheaf realization of global basis for Bridgeland's localizing algebra. In particular, we prove that the dual of canonical basis elements are part of our basis up to powers of $v$.Comment: In the new version, we add some content including: in section 6, we refine our construction and complete the perverse sheaves realization of Bridgeland's Hall algebra which is essentially isomorphic to the integral form of the whole quantum group; in section 7, we obtain new results about the global basis; in section 8, we compare our basis with Lusztig's (dual) canonical basi