In this paper, we follow an idea of Lusztig to define the Fourier transform on the Ringel–Hall algebra of a valued quiver (given by a quiver with automorphism). As an application, this provides a direct proof of the fact that the Ringel–Hall algebra of a valued quiver is independent of its orientation. Furthermore, by combining the BGP-reflection operators defined on double Ringel–Hall algebras of valued quivers with Fourier transforms, we obtain an alternative construction of Lusztig’s symmetries of the associated quantum enveloping algebras. This generalizes a result of Sevenhant and Van den Bergh in the quiver case.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.