Abstract
The preprojective algebra and the trivial extension algebra of a Dynkin quiver (in bipartite orientation) are very close to being a Koszul dual pair of algebras. In this case the usual duality theory may be adapted to show that each algebra has a periodic bimodule resolution built using the other algebra and some extra data: an algebra automorphism. A general theory of such ‘almost Koszul’ algebras is developed and other examples are given.
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Brenner, S., Butler, M.C.R. & King, A.D. Periodic Algebras which are Almost Koszul. Algebras and Representation Theory 5, 331–368 (2002). https://doi.org/10.1023/A:1020146502185
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DOI: https://doi.org/10.1023/A:1020146502185