On the structure of a Turaev coalgebra

Let pi be a discrete group with neutral element 1. We prove that a Turaev coalgebra H = {H-alpha}(alpha is an element of pi) is isomorphic to Radford's biproduct of a Turaev-Hopf subalgebra and a braided Turaev coalgebra if and only if the neutral component H-1 is isomorphic to some Radford's biproduct. This result would provide us a method to construct some new examples of non-semisimple Turaev coalgebras, and a possibility of the classifications of Turaev coalgebras by their Hopf subalgebras.