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. 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.

Chekanov-Eliashberg $\mathrm{dg}$-algebras for singular Legendrians

“…This finishes the proof □ Remark 2.5. The dg-algebra Π n (kQ, kF ) was also defined in [KW21] under the name relative derived preprojective algebra.…”

Relative Calabi–Yau structures

“…As we shall see in the next subsection, our examples of relative non-commutative orientations are induced by relative left Calabi-Yau structures in the sense of [BD19], [BD21]. For an alternative introduction to relative Calabi-Yau structures, see [KW21]. We now briefly review such structures, following the treatment in [BD21] and explain how they induce relative non-commutative orientations.…”

“…For details on this particular example, see [KW21]. More generally, a non-commutative relative co-orientation of dimension d on a dualizable functor f :…”