Abstract
We give a simple proof, using Auslander-Reiten theory, that the preprojective algebra of a basic hereditary algebra of finite representation type is Frobenius. We then describe its Nakayama automorphism, which is induced by the Nakayama functor on the module category of our hereditary algebra.
Highlights
The preprojective algebra is defined using only the module category, and we see it as a virtue that the proof stays within this setting
One might wonder if this automorphism has any interpretation in terms of the representation theory of the original quiver: can we describe the Nakayama automorphism for the Baer-Geigle-Lenzing preprojective algebra? Our second main result does this
The Nakayama automorphism of the preprojective algebra is induced by the Nakayama functor of the representation-finite hereditary algebra
Summary
The second is a result proved by both Platzeck-Auslander and Gabriel which characterizes representation finite hereditary algebras in terms of whether their injective modules are preprojective. This result was crucial in the proofs of self-injectivity by Brenner-Butler-King and Iyama-Oppermann. The Nakayama automorphism of the preprojective algebra is induced by the Nakayama functor of the representation-finite hereditary algebra. The above theorem is saying something different: the Nakayama functor for the hereditary algebra induces the Nakayama automorphism for the preprojective algebra This result appears to be new, though I suspect that it will not surprise the experts. This is both to simplify the exposition and to demonstrate the idea that, at least in some cases, we have a good understanding of a property of hereditary algebras precisely when our explanations work for d-hereditary algebras
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