A note on the order of the antipode of a pointed Hopf algebra

Let k be a field and let H denote a pointed Hopf k-algebra with antipode S. We are interested in determining the order of S. Building on the work done by Taft and Wilson in [7], we define an invariant for H, denoted mH, and prove that the value of this invariant is connected to the order of S. In the case where char k = 0, it is shown that if S has finite order then it is either the identity or has order 2 mH. If in addition H is assumed to be coradically graded, it is shown that the order of S is finite if and only if mH is finite. We also consider the case where char k = p > 0, generalizing the results of [7] to the infinite-dimensional setting