Derived Hall Algebras

Details Derived Hall Algebras Derived Hall Algebras

2005-01-21

arxiv.org/abs/math/0501343v4

Mathematics

Quantum Algebra

24 pages. A description of the derived Hall algebra of an hereditary abelian category has been added. Final version, to appear

Scientific paper

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.

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