Minimal models of loop spaces and suspensions (Q1313522)

This paper gives the solution to the 2 following problems: (1) Let \(X\) be a space. Determine the Sullivan minimal model for the loop multiplication \(\mu_ x : \Omega X \times \Omega X \to \Omega X\) and the Quillen model for the suspension comultiplication \(\nabla X : \Sigma X \to \Sigma X \vee \Sigma X\). (2) Let \(\psi : A \to A \otimes A\) be an associative comultiplication of a graded commutative algebra and let \(\varphi : L \to L \sqcup L\) be an associative comultiplication of a 1-connected graded Lie algebra \(L\); describe spaces \(X\) and \(Y\) such that \(\psi\) is \((\mu_ x)_ *\) and \(\varphi\) is \((\nabla Y)_ *\).