On two Hall algebra approaches to odd periodic triangulated categories (Q1705741)

The Hall algebra was firstly introduced by Steinitz around 1901 in the study of abelian \(p\)-groups. At the beginning, Hall algebras were closely related to the algebra of partitions and symmetric functions. The importance of Hall algebras introduced by Steinitz E.\ around 1901 has been highly recognized ever since the ground-breaking discovery of \textit{C. M. Ringel} [Invent. Math. 101, No. 3, 583--591 (1990; Zbl 0735.16009)]. Generally, the Hall algebra is a tool to encode the extension structure of a finitary category. Given an odd-periodic algebraic triangulated category, the authors compare Bridgeland's Hall algebra in the sense of \textit{T. Bridgeland} [Ann. Math. (2) 177, No. 2, 739--759 (2013; Zbl 1268.16017)] and the derived Hall algebra in the sense of \textit{B. Toën} [Duke Math. J. 135, No. 3, 587--615 (2006; Zbl 1117.18011)], \textit{J. Xiao} and \textit{F. Xu} [Duke Math. J. 143, No. 2, 357--373 (2008; Zbl 1168.18006)], and \textit{F. Xu} and \textit{X. Chen} [Algebr. Represent. Theory 16, No. 3, 673--687 (2013; Zbl 1275.16018)]. Then, it is shown that the former one is the twisted form of the tensor product of the latter one and a suitable group algebra.