QUADRATIC AND PINCZON ALGEBRAS

Didier Arnal (1) , Wissem Bakbrahem (2) , Mohamed Selmi (2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 LMPFSA - Laboratoire de Mathématique Physique, Fonctions Spéciales et Applications

Given a symmetric non degenerated bilinear form b on a vector space V , G. Pinczon and R. Ushirobira defined a bracket { , } on the space of multilinear skewsymmetric forms on V. With this bracket, the quadratic Lie algebra structure equation on (V, b) becomes simply {Ω, Ω} = 0. We characterize similarly quadratic associative, commutative or pre-Lie structures on (V, b) by the same equation {Ω, Ω} = 0, but on different spaces of forms. These definitions extend to quadratic up to homotopy algebras and allows to describe the corresponding cohomologies.

Mots clés: Graded Lie algebras; Cohomology; Quadratic algebras

Domaines: Algèbres quantiques [math.QA]

Didier Arnal, Wissem Bakbrahem, Mohamed Selmi. QUADRATIC AND PINCZON ALGEBRAS. 2016. ⟨hal-01280499⟩