Some remarks on triangulated categories and Kac–Moody algebras

In [L. Peng, J. Xiao, Triangulated categories and Kac–Moody algebras, Invent. Math. 140 (3) (2000) 563–603], the authors proved a theorem for constructing Hall type Lie algebras associated to 2-periodic triangulated categories. In particular, this theorem gives a global realization of symmetrizable Kac–Moody algebras. The purpose of this paper is to provide a refinement (meanwhile, a new proof) of the theorem by applying the approach of derived Hall algebras arising in homologically finite triangulated categories appeared in [B. Toën, Derived Hall algebras, Duke Math. J. 135 (3) (2006) 587–615] and [J. Xiao, F. Xu, Hall algebras associated to triangulated categories, Duke Math. J. 143 (2) (2008) 357–373].