Quadratic and pinczon algebras

TL;DR

This paper explores algebraic structures on vector spaces with symmetric bilinear forms, extending the Pinczon-Ushirobira bracket to characterize quadratic associative, commutative, and pre-Lie algebras, and their cohomologies.

Contribution

It generalizes the Pinczon-Ushirobira bracket to characterize various quadratic algebraic structures and extends these concepts to homotopy algebras and their cohomologies.

Findings

Characterization of quadratic associative, commutative, and pre-Lie structures via the bracket.

Extension of definitions to quadratic up to homotopy algebras.

Description of the cohomologies associated with these algebraic structures.

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.