Tensor products of homotopy Gerstenhaber algebras (Q650991)

Let \(A\) be an associative differential graded algebra. Let \(\bar{B}(A)\) be the normalized reduced bar complex of \(A\). The structure defined by a twisting cochain \(E: \bar{B}(A)\otimes\bar{B}(A)\rightarrow A\) yielding a (possibly non-associative) multiplication on the bar complex \(\bar{B} A\) is called a Hirsch algebra. The author considers Hirsch algebras such that the twisting cochains \(E\) vanish, \(E(a\otimes b) = 0\), when \(a = [a_1|\dots|a_p]\) belongs to a component of degree \(p>1\) of the bar complex \(\bar{B}(A)\). He uses the expression of level 3 Hirsch algebras to refer to this class of Hirsch algebras. The author proves, by an explicit inductive construction, that a tensor product \(A'\otimes A''\) of level 3 Hirsch algebras \(A'\) and \(A''\) inherits a Hirsch algebra structure, so that \(\bar{B}(A'\otimes A'')\) has a natural multiplication, and the classical shuffle map \(\bar{B}(A')\otimes\bar{B}(A'')\rightarrow\bar{B}(A'\otimes A'')\) is multiplicative.