Derived Hall Algebras

TL;DR

This paper introduces a new algebraic structure called the derived Hall algebra for dg-categories, proving its associativity, unitality, and its relation to classical Hall algebras, with explicit formulas and examples.

Contribution

It defines the derived Hall algebra for dg-categories, proves its fundamental properties, and relates it to classical Hall algebras with explicit formulas and examples.

Findings

Derived Hall algebra is associative and unital.

Contains the classical Hall algebra for abelian categories.

Provides explicit formulas for derived Hall numbers.

Abstract

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.