Superpotentials and higher order derivations

Bocklandt, R., Schedler, T. and Wemyss, M. (2010) Superpotentials and higher order derivations. Journal of Pure and Applied Algebra, 214(9), pp. 1501-1522. (doi: 10.1016/j.jpaa.2009.07.013)

Abstract

We consider algebras defined from quivers with relations that are kth order derivations of a superpotential, generalizing results of Dubois-Violette to the quiver case. We give a construction compatible with Morita equivalence, and show that many important algebras arise in this way, including McKay correspondence algebras for View the MathML source for all n, and four-dimensional Sklyanin algebras. More generally, we show that any N-Koszul, (twisted) Calabi–Yau algebra must have a (twisted) superpotential, and construct its minimal resolution in terms of derivations of the (twisted) superpotential. This yields an equivalence between N-Koszul twisted Calabi–Yau algebras A and algebras defined by a superpotential ω such that an associated complex is a bimodule resolution of A. Finally, we apply these results to give a description of the moduli space of four-dimensional Sklyanin algebras using the Weil representation of an extension of View the MathML source.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Wemyss, Professor Michael
• Authors: Bocklandt, R., Schedler, T., and Wemyss, M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Pure and Applied Algebra
• Publisher: Elsevier
• ISSN: 0022-4049
• ISSN (Online): 1873-1376
• Published Online: 01 February 2010
• Copyright Holders: Copyright © 2010 Elsevier B.V.
• First Published: First published in Journal of Pure and Applied Algebra 214(9):1501-1522
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher