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Pré-Publication, Document De Travail Année : 2024

RANK TWO ARTIN-SCHELTER REGULAR ALGEBRAS AND NON COMMUTING DERIVATIONS

Vincent Beck
César Lecoutre
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Résumé

If ∆ and Γ are two derivations of a commutative algebra A such that ∆Γ-Γ∆ = ∆ is locally nilpotent, one can endow A with a new product * whose filtered semiclassical limit is the Poisson structure ∆ ∧ Γ. In this article we first study theses (Poisson) algebras from an algebraic point of view, and when A is a polynomial algebra, we investigate their homological properties. In particular, if the derivations ∆ and Γ are linear, we prove that the algebras (A, *) are Artin-Schelter regular algebras. Assuming furthermore a technical condition on Γ, we show that the algebra (A, *) is Calabi-Yau if and only if the trace of Γ is equal to 1 if and only if the Poisson algebra (A, ∆ ∧ Γ) is unimodular.
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Dates et versions

hal-04458688 , version 1 (15-02-2024)
hal-04458688 , version 2 (26-02-2024)

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Paternité - Pas d'utilisation commerciale - Partage selon les Conditions Initiales

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Vincent Beck, César Lecoutre. RANK TWO ARTIN-SCHELTER REGULAR ALGEBRAS AND NON COMMUTING DERIVATIONS. 2024. ⟨hal-04458688v2⟩
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