Derived Hall algebras.

Bertrand Toen (1)

1 LEP - Laboratoire Émile Picard

The purpose of this work is to define a derived Hall algebra $\mathcal{DH}(T)$, associated to any dg-category $T$ (under some finiteness conditions). Our main theorem states that $\mathcal{DH}(T)$ is associative and unital. It is shown that $\mathcal{DH}(T)$ contains the usual Hall algebra $\mathcal{H}(T)$ when $T$ is an abelian category. We will also prove an explicit formula for the derived Hall numbers purely in terms of invariants of the triangulated category associated to $T$. As an example, we describe the derived Hall algebra of an hereditary abelian category.

Domaines: Algèbres quantiques [math.QA]; Topologie algébrique [math.AT]

Bertrand Toen. Derived Hall algebras.. Duke Mathematical Journal, 2006, 135 (3), pp.587-615. ⟨10.1215/S0012-7094-06-13536-6⟩. ⟨hal-00772977⟩