On root categories of finite-dimensional algebras

For any finite-dimensional algebra A over a field k of finite global dimension, we investigate the root category RA as the triangulated hull of the 2-periodic orbit category of A via the construction of B. Keller in “On triangulated orbit categories”. This is motivated by Ringel–Hall Lie algebras associated to 2-periodic triangulated categories. As an application, we study the Ringel–Hall Lie algebras for a class of finite-dimensional k-algebras of global dimension 2, which turns out to give an alternative answer to a question of GIM-Lie algebras by Slodowy in “Beyond Kac–Moody algebra, and inside”.