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 $\mathrm{mod}(\mathrm{End}(T{)}^{op})$.

arXiv comment53 pages, minor changes, accepted by Adv. Math

References in corpus (2)

• A Double Hall Algebra Approach to Affine Quantum Schur--Weyl Theory
• Semi-derived Hall algebras and tilting invariance of Bridgeland-Hall algebras

Cited by in corpus (6)

• Hall algebras and quantum symmetric pairs I: foundations
• $ı$Hall algebra of the projective line and $q$-Onsager algebra
• Modified Ringel-Hall Algebras, Green's formula and Derived Hall Algebras
• $ı$Hall algebra of Jordan quiver and $ı$Hall-Littlewood functions
• $ı$quantum groups of split type via derived Hall algebras
• Quantum Borcherds-Bozec algebras via semi-derived Ringel-Hall algebras