Open AccessFourier transforms and Ringel–Hall algebras of valued quivers
Author(s) -
Chenyang Ma
Publication year - 2021
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/proc/15403
Subject(s) - quiver , homogeneous space , pure mathematics , mathematics , fourier transform , algebra over a field , automorphism , mathematical analysis , geometry
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.