2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/164743We 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 homotopyQuantum Algebra18G55,16W30,17B63, 16E45text