  In our previous paper math/0502157 we classified a large class of
finite-dimensional pointed Hopf algebras up to isomorphism. However the
following problem was left open for Hopf algebras of of type $A,D$ or $E_6$,
that is whose Cartan matrix is connected and allows a non-trivial automorphism
of the corresponding Dynkin diagram. In this case we described the isomorphisms
between two such Hopf algebras with the same Cartan matrix only implicitly. The
problem is whether it is possible to compute the isomorphisms in terms of the
defining families of parameters. In the present paper we solve this problem for
type $A$.
