On bihom-gerstenhaber algebras up to homotopy

In the current research paper, we define and investigate the concept of BiHom-Gerstenhaber algebra. This algebraic structure is defined by two linear maps f, g, a BiHom-commutative law \(''\wedge ''\) $_{}^{\prime \prime }{\wedge }^{\prime \prime }$ , a BiHom-Lie law \(''[,\,]''\) $_{}^{\prime \prime }{[,]}^{\prime \prime }$ with degrees 0 and \(-1\) $-1$ respectively over a graded vector space \(\mathfrak {g}\) $\mathfrak{g}$ . Additionally, it satisfies a compatibility condition called BiHom-Leibniz relation. Furthermore, we will provide an explicit construction of the associated BiHom-Gerstenhaber algebras up to homotopy. More precisely, this algebraic structure is defined by a structure of BiHom-cocommutative coalgebra \(\big (C,\Delta _{(f,g)},f,g\big )\) $(C,{\mathrm{\Delta }}_{(f,g)},f,g)$ , a structure of BiHom-coLie coalgebra \(\big (C,\delta _{(f,g)},f,g\big )\) $(C,{\delta }_{(f,g)},f,g)$ . It is equipped with a coderivation Q for two coproducts \(\Delta _{(f,g)}\) ${\mathrm{\Delta }}_{(f,g)}$ and \(\delta _{(f,g)}\) ${\delta }_{(f,g)}$ with degree 0 verifying \(Q^2=0\) $Q^2=0$ and a compatibility relations, called BiHom-coLeibniz relations. This bicoalgebra is also called BiHom-Gerstenhaber algebra up to homotopy or BiHom- \(\mathfrak {g}_{\infty }\) ${\mathfrak{g}}_{\infty }$ -algebra.