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 $$ with $n\geq 3$ have a finite order (whenever defined). However, if $\Bbbk$ is a field of characteristic zero, $\langle x\rangle ^{n}$ is defined and vanishes in $H(A\otimes \Bbbk )$ for all $n$. If $p$ is an odd prime, the Kraines formula $\langle x\rangle ^{p}=-\beta \mathcal{P}_{1}(x)$ lifts to $H^{\ast }(A\otimes {\mathbb{Z}}_{p}).$ Applications of the existence of polynomial generators in the loop homology and the Hochschild cohomology with a $G$-algebra structure are given.