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

TL;DR

This paper introduces the concept of pre-(a,b)-algebras up to homotopy, providing explicit constructions for these structures, which generalize pre-Gerstenhaber and pre-Poisson algebras, expanding the theoretical framework of algebraic structures with two operations.

Contribution

It defines pre-(a,b)-algebras up to homotopy and offers explicit constructions, extending the theory of algebraic structures with two operations.

Findings

Explicit construction of pre-(a,b)-algebras up to homotopy

Generalization of pre-Gerstenhaber and pre-Poisson algebras

Framework for algebraic structures with two operations

Abstract

We study in this article the concept of algebra up to homotopy for a structure defined by two operations . Important examples of such structure are those of pre-Gerstenhaber and pre-Poisson algebras. Given a structure of pre-commutative and pre-Lie algebra for two shifts of degree given by a and b, we define the structure of a pre-(a, b)-algebra and we give an explicit construction of the associated algebra up to homotopy.