Preprojective algebras and generalisations: A short survey

The preprojective algebra of a hereditary algebra $H$ can be defined as a certain orbit construction of the regular representation generated by the Auslander-Reiten translation. In this short survey, we will look at two important generalisations, namely, the contracted preprojective algebra and the total preprojective algebra. We will include several open problems and questions motivated by examples in the hope to stimulate future research on general orbit algebras of $H$-modules.