# Koszul calculus of preprojective algebras

1 AGL - Algèbre, géométrie, logique
ICJ - Institut Camille Jordan [Villeurbanne]
Abstract : We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A$_1$ and A$_2$, vanishes in any (co)homological degree $p>2$. Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to its (higher) homological calculus, by exchanging degrees $p$ and $2-p$, and we prove a generalised version of the 2-Calabi-Yau property. For the ADE Dynkin graphs, the preprojective algebras are not Koszul and they are not Calabi-Yau in the sense of Ginzburg's definition, but they satisfy our generalised Calabi-Yau property and we say that they are Koszul complex Calabi-Yau (Kc-Calabi-Yau) of dimension $2$. For Kc-Calabi-Yau (quadratic) algebras of any dimension, defined in terms of derived categories, we prove a Poincar\'e Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.
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Cited literature [48 references]

https://hal.archives-ouvertes.fr/hal-02132927
Contributor : Rachel Taillefer <>
Submitted on : Tuesday, March 17, 2020 - 2:44:14 PM
Last modification on : Tuesday, March 24, 2020 - 10:04:22 AM

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### Identifiers

• HAL Id : hal-02132927, version 2
• ARXIV : 1905.07906

### Citation

Roland Berger, Rachel Taillefer. Koszul calculus of preprojective algebras. 2020. ⟨hal-02132927v2⟩

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