An introduction to relative Calabi–Yau structures

Abstract

These are notes taken by the second author for a series of three lectures by the first author on absolute and relative Calabi–Yau completions and Calabi–Yau structures given at the workshop of the International Conference on Representations of Algebras which was held online in November 2020. After a quick reminder on dg categories and their Hochschild and cyclic homologies, we present examples of absolute and relative Calabi–Yau completions (in the sense of Yeung) relevant for (higher) representation theory of finite-dimensional algebras. We conclude with the definition of relative (left and right) Calabi–Yau structures after Brav–Dyckerhoff.