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 AA. When x∈H(A)x∈H(A) with ZZ coefficients and x2=0x2=0, the symmetric Massey products 〈x〉n〈x〉n with n≥3n≥3 have a finite order (whenever defined). However, if kk is a field of characteristic zero, 〈x〉n〈x〉n is defined and vanishes in H(A⊗k)H(A⊗k) for all nn. If pp is an odd prime, the Kraines formula 〈x〉p=−βP1(x)〈x〉p=−βP1(x) lifts to H∗(A⊗Zp)H∗(A⊗Zp). Applications of the existence of polynomial generators in the loop homology and the Hochschild cohomology with a GG-algebra structure are given.