Frobenius rational loop algebra

Recently R. Cohen and V. Godin have proved that the loop homology H*+m (LM; k) of a closed oriented m dimensional manifold M with coefficients in a field k has the structure of a unital Frobenius algebra without counit. When the characteristic of k is zero we describe explicitly the dual of the coproduct H*(LM; k)(circle times 2) -> H*+m(LM; k) and prove that it respects the Hodge decomposition of H*( LM; k).