Lie algebras and vn-periodic spaces

We consider a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in vn -periodic homotopy groups. The case n = 0 corresponds to rational homotopy theory. In analogy with Quillen's results in the rational case, we prove that this vn-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in T(n)-local spectra. We also compare it to the homotopy theory of commutative coalgebras in T(n)-local spectra, where it turns out there is only an equivalence up to a certain convergence issue of the Goodwillie tower of the identity.