Lie algebras and $v_n$-periodic spaces

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