Massey Product and Twisted Cohomology of A-infinity Algebras

We study the twisted cohomology groups of $A_{\mathrm{\infty }}$-algebras defined by twisting elements and their behavior under morphisms and homotopies using the bar construction. We define higher Massey products on the cohomology groups of general $A_{\mathrm{\infty }}$-algebras and establish the naturality under morphisms and their dependency on defining systems. The above constructions are also considered for $C_{\mathrm{\infty }}$-algebras. We construct a spectral sequence converging to the twisted cohomology groups an show that the higher differentials are given by the $A_{\mathrm{\infty }}$-algebraic Massey products.