Year: 2020
Type: article
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 $d$ -hereditary, all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a $d$ -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.
Source: Compositio Mathematica
Authors: Joseph Grant, Osamu Iyama
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Cluster-tilting theory, G4776928572, Existence of higher dimensional crepant resolutions and generlization of the McKay correspondence, Research of lattices, automorphic forms and moduli spaces, New development of representation theory of orders, Comprehensive research on representation theory of orders
OpenAlex ID: W2916990581