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.
Original language  English 

Pages (fromto)  25882627 
Number of pages  40 
Journal  Compositio Mathematica 
Volume  156 
Issue number  12 
DOIs  
Publication status  Published  1 Dec 2020 
Keywords
 CalabiYau algebra
 Jacobi algebra
 periodic resolution
 preprojective algebra
 superpotential
Profiles

Joseph Grant
 School of Mathematics  Lecturer in Pure Mathematics
 Algebra and Combinatorics  Member
Person: Research Group Member, Academic, Teaching & Research