Sheaf realization of Bridgeland's Hall algebra of Dynkin type

Abstract

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.

In the new version, we add a new result (Theorem 8.1): the $\mathbb{Z}[v,v^{-1}]$-algebra $\mathcal{D}{\mathcal{K}}^{ss,\ast ,\mathrm{red}}$ constructed by semisimple complexes and the restriction functor in section 6, is isomorphic to the extension counting integral form of the generic Bridgeland's Hall algebra and is isomorphic to the Poisson integral form of the quantum group