Internally Calabi-Yau algebras and cluster-tilting objects

TL;DR

This paper introduces the concept of internally Calabi-Yau algebras relative to an idempotent, linking them to cluster-tilting objects in Frobenius categories and exploring their properties, especially in the context of frozen Jacobian algebras.

Contribution

It defines internally Calabi-Yau algebras with respect to an idempotent and connects them to endomorphism algebras of cluster-tilting objects, providing a new approach to model cluster algebras with frozen variables.

Findings

Internally Calabi-Yau algebras relate to endomorphism algebras of cluster-tilting objects.

A bimodule internally Calabi-Yau condition implies the Frobenius category is stably Calabi-Yau.

A candidate bimodule resolution for frozen Jacobian algebras is proposed, linking to their Calabi-Yau properties.

Abstract

We describe what it means for an algebra to be internally d-Calabi-Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1)-cluster-tilting objects in certain stably (d-1)-Calabi-Yau Frobenius categories, as observed by Keller-Reiten. We show that an internally d-Calabi-Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d-1)-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger 'bimodule' internally d-Calabi-Yau condition, this Frobenius category is stably (d-1)-Calabi-Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi-Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot's construction in the coefficient-free case.