Semi-derived Ringel-Hall algebras and Drinfeld double

Abstract

Let $\mathcal{A}$ be an arbitrary hereditary abelian category that may not have enough projective objects. For example, $\mathcal{A}$ can be the category of finite-dimensional representations of a quiver or the category of coherent sheaves on a smooth projective curve or on a weighted projective line. Inspired by the works of Bridgeland and Gorsky, we define the semi-derived Ringel-Hall algebra of $\mathcal{A}$, denoted by $\mathcal{S}\mathcal{D}\mathcal{H}_{\mathbb{Z}/2}(\mathcal{A})$, to be the localization of a quotient algebra of the Ringel-Hall algebra of the category of $\mathbb{Z}/2$-graded complexes over $\mathcal{A}$. We obtain the following three main results. The semi-derived Ringel-Hall algebra has a natural basis. A twisted version of the semi-derived Ringel-Hall algebra of $\mathcal{A}$ is isomorphic to the Drinfeld double of the twisted extended Ringel-Hall algebra $\mathcal{H}_{tw}^e(\mathcal{A})$ of $\mathcal{A}$. If $\mathcal{A}$ has a tilting object $T$, then its semi-derived Ringel-Hall algebra is isomorphic to the $\mathbb{Z}/2$-graded semi-derived Hall algebra $\mathcal{S}\mathcal{D}\mathcal{H}_{\mathbb{Z}/2}(\mathrm{add} T)$ of the exact category $\mathrm{add} T$ defined by Gorsky, and so is isomorphic to Bridgeland's Hall algebra of $\mod (\mathrm{End}(T)^{op})$.