Quadratic and pinczon algebras
  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 {\^a, \^a} = 0. We characterize similarly quadratic associative, commutative or pre-Lie structures on (V, b) by the same equation {\^a, \^a} = 0, but on different spaces of forms. These definitions extend to quadratic up to homotopy algebras and allows to describe the corresponding cohomologies. 
Didier Arnal (IMB), Wissem Bakbrahem (LMPFSA), Mohamed Selmi (LMPFSA)
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