Higher preprojective algebras, Koszul algebras, and superpotentials

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Grant, Joseph and Iyama, Osamu (2020) Higher preprojective algebras, Koszul algebras, and superpotentials. Compositio Mathematica, 156 (12). pp. 2588-2627. ISSN 0010-437X

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Abstract

In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case, we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is -hereditary, all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a -hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra.

Item Type:	Article
Additional Information:	E-pub ahead of print: 1 Feb 2021
Uncontrolled Keywords:	calabi-yau algebra,jacobi algebra,periodic resolution,preprojective algebra,superpotential,algebra and number theory ,/dk/atira/pure/subjectarea/asjc/2600/2602
Faculty \ School:	Faculty of Science > School of Mathematics (former - to 2024)
UEA Research Groups:	Faculty of Science > Research Groups > Algebra and Combinatorics (former - to 2024) Faculty of Science > Research Groups > Algebra, Number Theory, Logic, and Representations (ANTLR)
Related URLs:	http://www.scopus.com/inward/record.url?... https://arxiv.org/abs/1902.07878 https://arxiv.org/abs/1902.07878
Depositing User:	LivePure Connector
Date Deposited:	19 May 2020 00:20
Last Modified:	01 Jul 2025 00:38
URI:	https://ueaeprints.uea.ac.uk/id/eprint/75213
DOI:	10.1112/S0010437X20007538

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