Filtered Hirsch Algebras

Motivated by the cohomology theory of loop spaces, we consider a special class of higher order homotopy commutative differential graded algebras and construct the filtered Hirsch model for such an algebra $A$. When $x\in H(A)$ with $\mathbb{Z}$ coefficients and $x^2=0,$ the symmetric Massey products $% \langle x\rangle ^{n}$with$n\geq 3$ have a finite order (whenever defined). However, if $\mathbb{k}$ is a field of characteristic zero, $⟨x{⟩}^n$ is defined and vanishes in $H(A\otimes \mathbb{k})$ for all $n$. If $p$ is an odd prime, the Kraines formula $⟨x{⟩}^p=-\beta {\mathcal{P}}_1(x)$ lifts to $H^{\ast }(A\otimes {\mathbb{Z}}_p)\mathrm{.}$ Applications of the existence of polynomial generators in the loop homology and the Hochschild cohomology with a $G$-algebra structure are given.