On antipodes in pointed Hopf algebras

Abstract

If S is the antipode of a Hopf algebra H, the order of S is defined to be the smallest positive integer n such that S n = I (in case such integers exist) or ∞ (if no such integers exist). Although in most familiar examples of Hopf algebras the antipode has order 1 or 2, examples are known of infinite dimensional Hopf algebras in which the antipode has infinite order or arbitrary even order [1, 4, 6] and also of finite dimensional Hopf algebras in which the antipode has arbitrary even order [3, 5]. Some sufficient conditions for the antipode to have order ⩽4 are known [2, 4], but the following questions remain open: Does the antipode of a finite dimensional Hopf algebra necessarily have finite order? If the antipode S of a Hopf algebra H has finite order is that order bounded by some function of dim H? In this paper, by constructing a certain basis for an arbitrary pointed coalgebra and studying the action of the antipode on the elements of such a basis for a pointed Hopf algebra, we obtain affirmative answers to the second question in case H is pointed and to the first question in case H is pointed over a field of prime characteristic. We use freely the definitions, notation, and results of [4].