Hopf algebras over number rings

Abstract

Sweedler [7, p. 891 first mentioned that the antipodes of Hopf algebras over fields may have any even order or infinite order. The examples he gave are infinite dimensional. Radford [S] proved later that the order of the antipode of a finite dimensional Hopf algebra over a field must be finite. A sequence of papers published by Taft, Wilson, and Radford [6,9, lo], showed that it is possible to construct finite dimensional Hopf algebras over any field with antipodes of any given even order. Waterhouse [ 111 generalized the finiteness of antipodal order to finitely generated projective Hopf algebras over any commutative ring. So, there is a natural question of whether we could construct Hopf algebras whose antipodal orders are arbitrary even integers at the level of commutative rings. In this paper we give positive answers for number rings Z [ l/n]. Originally, Taft [S] introduced examples of finite dimensional Hopf algebras with antipodes of arbitrary even order 2n over any field that contains a primitive nth root of unity. For fields of characteristic 0 or p not dividing n this restriction was dropped later in a joint work by him, Radford, and Wilson [6]. (Note: for general fields, see [lo].) The main ideas there are using a finite Galois extension of the base field to recover all primitive nth roots, constructing one copy of the Hopf algebra for each root, tensoring these together, defining a group of semilinear automorphisms with respect to the Galois group, taking the set of fixed elements, and obtaining the required Hopf algebra as a form. These two results are restated in Section 2. In Section 2 we present basic notations and conventions used in later discussion. Some elementary results following from these notions are quickly pointed out.