Module derivations and the adjoint action of a finite loop space (Q1125248)

Let \(G\) be a loop space with the homotopy type of a finite complex. The natural adjunction map \(\text{Ad}:G\times\Omega G\to\Omega G\) gives the space \(\Omega G\times G\) a group structure which is isomorphic to the group \(LG\) of the free loops on \(G\). The aim of the paper is to give another proof of a recent result of Iwase and Kono: If \(G\) is simply connected then the following three conditions are equivalent: (i) \(H^*(G;Z)\) has no 2-torsion. (ii) \(H^*(\text{(Ad}; \mathbb{Z}/_2)= H^*(\text{pr}_2; \mathbb{Z}/_2)\) where \(\text{pr}_2\), denotes the projection of the second factor. (iii) There is an isomorphism of \(H^*(\text{BG};\mathbb{Z}/_2)\)-algebra \[ H^*(\text{BLG};\mathbb{Z}/_2)\cong H^*(\text{BG};\mathbb{Z}/_2) \otimes H^*(G; \mathbb{Z}/_2) \] which respects the inclusion onto the second factor.