An introduction to relative Calabi-Yau structures

TL;DR

This paper introduces the concept of relative Calabi-Yau structures, exploring their definitions, examples, and relevance to higher representation theory and categorification, with connections to Hochschild homology and preprojective algebras.

Contribution

It provides a comprehensive introduction to relative Calabi-Yau structures, including definitions, examples, and their applications in representation theory and categorification.

Findings

Examples relate to higher preprojective algebras

Connections established with Hochschild and cyclic homologies

Definitions of relative Calabi-Yau structures presented

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. Such structures are relevant for (higher) representation theory as well as for the categorification of cluster algebras with coefficients. 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). In many examples, these are related to higher preprojective algebras in the sense of Iyama-Oppermann. We conclude with the definition of relative (left and right) Calabi-Yau structures after Brav-Dyckerhoff.