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Pré-Publication, Document De Travail Année : 2019

Koszul calculus of preprojective algebras

Résumé

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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Dates et versions

hal-02132927 , version 1 (17-05-2019)
hal-02132927 , version 2 (17-03-2020)

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Roland Berger, Rachel Taillefer. Koszul calculus of preprojective algebras. 2019. ⟨hal-02132927v1⟩
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