Remarks on Hall algebras of triangulated categories

Abstract

In this paper, we introduce a motivic version of Toën’s derived Hall algebra. Then we point out that the two kinds of Hall algebras in the sense of Toën and Kontsevich–Soibelman, respectively, are Drinfeld dual pairs, not only in the classical case (by counting over finite fields) but also in the motivic version. Consequently they are canonically isomorphic. All proofs, including that for the most important associative property, are deduced in a self-contained way by analyzing the symmetry properties around the octahedral axiom, a method we used previously.