Quadratic hom-right symmetric algebras

Abstract

Given a symmetric non-degenerate bilinear form b on a vector space g, G. Pinczon and R. Ushirobira have defined a bracket {,} on the space of multilinear skewsymmetric forms on g. With this bracket, the quadratic Lie algebra structure equation on (g,b) becomes simply {Ω,Ω}=0. Following the same program, we characterize the quadratic hom-right symmetric structures on (g,b) by the same equation {Ω,Ω}=0, on the space of ‘bi-symmetric’ forms. This characterization extends to quadratic hom-right symmetric algebras up to homotopy and allows us to describe the corresponding cohomology.