Internally Calabi–Yau algebras and cluster-tilting objects

Pressland, M. (2017) Internally Calabi–Yau algebras and cluster-tilting objects. Mathematische Zeitschrift, 287(1-2), pp. 555-585. (doi: 10.1007/s00209-016-1837-0)

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)-clustertilting 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Pressland, Dr Matthew
• Authors: Pressland, M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Mathematische Zeitschrift
• Publisher: Springer
• ISSN: 0025-5874
• ISSN (Online): 1432-1823
• Published Online: 03 January 2017
• Copyright Holders: Copyright © 2017 The Author
• First Published: First published in Mathematische Zeitschrift 287(1-2): 555-585
• Publisher Policy: Reproduced under a Creative Commons License