Remarks on Hall algebras of triangulated categories
Jie Xiao, Fan Xu
TL;DR
This paper explores the duality between different types of Hall algebras in triangulated categories, establishing new connections and constructing motivic analogues that unify existing algebraic frameworks.
Contribution
It introduces the Drinfeld dual concept for algebras, shows the duality between Kontsevich-Soibelman Hall algebras and derived Hall algebras, and constructs a motivic analogue linking these structures.
Findings
Hall algebras are Drinfeld duals of derived Hall algebras
Motivic derived Hall algebra is isomorphic to the motivic Hall algebra
Unifies different algebraic structures in triangulated categories
Abstract
We introduce the notion of the Drinfeld dual of an algebra and show that Hall algebras defined by Kontsevich-Soibelman in \cite{KS} are the Drinfeld duals of derived Hall algebras defined in \cite{Toen2005} and \cite{XX2006}. Moreover, we construct the motivic analogue of a derived Hall algebra and prove that it is isomorphic to the motivic Hall algebra constructed in \cite{KS}.
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