Quadratic and pinczon algebras

Abstract

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.