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 {â, â} = 0. We characterize similarly quadratic associative, commutative or pre-Lie structures on (V, b) by the same equation {â, â} = 0, but on different spaces of forms. These definitions extend to quadratic up to homotopy algebras and allows to describe the corresponding cohomologies.

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