A lecture note on covering theory in representation theory of algebras
Yuming Liu, Nengqun Li, Bohan Xing, Pengyun Chen
TL;DR
This note provides an elementary, cohesive exposition of covering theory in representation theory of algebras, connecting coverings of quivers and quivers with relations to Galois actions, orbit categories, and universal covers. It develops push-down and pull-up functors, their adjunction, and shows how these tools transport AR-structure, representation type, and derived-category information across coverings, including graded and non-skeletal settings. A central theme is the use of covers (especially universal covers) to detect finite, tame, and wild representation types, and to describe indecomposables via strings and bands in string algebras. The work also extends the framework to general $k$-categories and derived categories, establishing precoverings and graded adjoint pairs that yield (i) AR-quiver correspondences, (ii) module and derived-category transfers, and (iii) connections to repetitive algebras and derived equivalences, while noting limitations and characteristic-dependent phenomena.
Abstract
Covering theory is an important tool in representation theory of algebras, however, the results and the proofs are scattered in the literature. We give an introduction to covering theory at a level as elementary as possible.