The Ringel-Hall Lie algebra of a spherical object (Q2880400)

Let \(\mathcal{C}\) be a triangulated category, and \(w\) an integer. An object \(S\) of \(\mathcal{C}\) is a \(w\)-spherical object if the graded endomorphism algebra \(\bigoplus_{p\in\mathbb Z}\mathrm{Hom}_{\mathcal{C}} (S,\Sigma^pS)\) is isomorphisc to \(\lambda=k[s]/s^2\), where \(s\) is of degree \(w\). Let \(S_w\) be the algebraic triangulated category generated by a \(w\)-spherical object \(S\). The Hall algebra of \(S_w\) was computed \textit{B. Keller}, \textit{D. Yang} and \textit{G. Zhou} [``The Hall algebra of a spherical object'', J. Lond. Math. Soc., II. Ser. 80, No. 3, 771--784 (2009; Zbl 1244.18008)]. The authors continue to study the Ringel-Hall algebra of a spherical in this paper.NEWLINENEWLINE To this aim, they first prove that each orbit category of \(S_w\) is triangulated (moreover, is triangle equivalent to a certain orbit category of the bounded derived category of a standard tube). Then, the authors obtain a description of the associated Ringel-Hall Lie algebra by applying Peng-Xiao's construction [\textit{L. Peng} and \textit{J. Xiao}, ``Triangulated categories and Kac-Moody algebras'', Invent. Math. 140, No. 3, 563--603 (2000 Zbl 0966.16006)].