ISOMORPHISM CLASSES AND AUTOMORPHISMS OF FINITE HOPF ALGEBRAS OF TYPE An NICOLÁS ANDRUSKIEWITSCH AND HANS-JÜRGEN SCHNEIDER

In [AS2] 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 E6, 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. To our surprise there are closed formulas for these isomorphisms. They are based on an action of the non-trivial automorphism σ of the Dynkin diagram on the parameter spaces of the Hopf algebras of type A.