Almost split sequences for symmetric non-semisimple Hopf algebras

We first prove that for a finite dimensional non-semisimple Hopf algebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-module X, we can construct a special kind of exact sequence ending with it. The main aim of this paper is to determine when this special exact sequence is an almost split one. For this aim, we restrict H to be unimodular and the square of its antipode to be an inner automorphism. As a special case, we give an application to the quantum double D(H)=(Hop)* H) of any non-semisimple Hopf algebra.