Les (a,b)-alg\`ebres \`a homotopie pr\`es

TL;DR

This paper introduces a new class of algebraic structures called (a,b)-algebras up to homotopy, generalizing existing algebraic frameworks by incorporating two degree shifts and explicit constructions.

Contribution

It defines (a,b)-algebras up to homotopy, extending G_infinity and P_infinity algebra concepts with explicit construction methods.

Findings

Defined (a,b)-algebras up to homotopy.

Extended G_infinity and P_infinity algebra structures.

Provided explicit construction for these generalized algebras.

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