The purpose of this work is to define a derived Hall algebra DH(T ), associated to any dg-category T (under some finiteness conditions), generalizing the Hall algebra of an abelian category. Our main theorem states that DH(T ) is associative and unital. When the associated triangulated category [T ] is endowed with a t-structure with heart A, it is shown that DH(T ) contains the usual Hall algebra H(A). 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.
In REF , (derived) Hall algebras of dg-categories have been introduced from a purely categorical approach.
Derived Hall algebras
Duke Mathematical Journal
Mathematics