Koszul calculus of preprojective algebras

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, the (higher) cohomological calculus is isomorphic as a bimodule to the (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 generalised Calabi-Yau. For generalised Calabi-Yau algebras of any dimension, defined in terms of derived categories, we prove a Poincaré Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.
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Contributor : Rachel Taillefer <>
Submitted on : Friday, May 17, 2019 - 4:46:46 PM
Last modification on : Thursday, June 13, 2019 - 11:39:17 AM


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  • HAL Id : hal-02132927, version 1
  • ARXIV : 1905.07906


Roland Berger, Rachel Taillefer. Koszul calculus of preprojective algebras. 2019. ⟨hal-02132927⟩



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