The antipode of a finite-dimensional Hopf algebra over a field has finite order

Abstract

The purpose of this note is to indicate a proof of the fact that the order of the antipode of a finite-dimensional Hopf algebra over a field has finite order. It has been shown [2], [5] that the order of the antipode of an infinite-dimensional Hopf algebra is not necessarily finite; and the order of the antipode is finite in the finite-dimensional case if the Hopf algebra is unimodular [1] or pointed and the ground field has prime characteristic [6] . Using the bilinear form introduced and studied in [3] we prove that the order is finite for any finite-dimensional Hopf algebra over a field. The bilinear form, integrals, grouplike elements, one-dimensional ideals, and the antipode are all related in rather intriguing ways. The proof of the finite order theorem is based on an explicit formula describing the fourth power of the antipode, as suggested by Theorem 5.5 of [1]. Full details will appear elsewhere [4].