Les (a,b)-algèbres à homotopie près

Abstract

We study in this article the concepts of algebra up to homotopy for a structure defined by two operations $ \pt $ and $[,]$. Having determined the structure of $G_{\infty }$ algebras and $P_{\infty }$ algebras, we generalize this construction and we define a structure of $(a,b)$-algebra up to homotopy. Given a structure of commutative and differential graded Lie algebra for two shifts degree given by $a$ and $b$, we will give an explicit construction of the associate algebra up to homotopy