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On Kaplansky's fifth conjecture

Yorck Sommerhäuser

Abstract

We prove that the antipode of a semisimple Hopf algebra is an involution if the characteristic of the base field is very large.

Introduction

In his Chicago lecture notes [5], I. Kaplansky set up a series of ten conjectures on Hopf algebras that he considered as important problems of this theory. Nearly all of these conjectures turned out to be puzzling as well as fundamental, and therefore have stimulated a lot of research in the area. Recently, important progress has been made on the first (cf. [17]), the sixth (cf. [18], [14]), the eighth (cf. [4], [35], [11]) and the tenth (cf. [31]) of these conjectures. The reader is referred to [16] and [30] for more precise information on the status of these conjectures. Closely related to the eighth conjecture is the classification problem for semisimple Hopf algebras, where A. Masuoka has contributed important results (cf. [12] and the references there).

Kaplansky's fifth conjecture states that the antipode of a semisimple Hopf algebra is an involution. This was proved by R. Larson and D. Radford in two closely related papers (cf. [7], [8]) in the case of a base field of characteristic zero. Their proof is carried out in two steps, the first one being to show that the Hopf algebra under consideration is also cosemisimple, the second one being to prove that the antipode of a semisimple and cosemisimple Hopf algebra is an involution. Their methods used for the second step were also powerful enough to prove Kaplansky's seventh conjecture. In the first step, their proof rests on the observation that a complex number times its conjugate yields a nonnegative real number, and therefore does not easily generalize to fields of positive characteristic. In the present paper, we improve on this step and give a proof of the conjecture in the case of a finite dimensional Hopf algebra over a field of large positive characteristic. More precisely, we prove that the antipode of a semisimple Hopf algebra is an involution if the characteristic p of the base field satisfies the inequality p > mm-4, where p = char K and m = 2 (dim H)2. Our techniques rely on the analysis of the structure of the character ring of H, as do the techniques used by G. I. Kac, Y. Zhu and M. Lorenz to prove Kaplansky's eighth conjecture in characteristic zero and the techniques used by W. D. Nichols and M. B. Richmond to prove results on Kaplansky's sixth conjecture.

The article is organized as follows: In Section 2, we discuss a technique to adjoin a grouplike element in such a way that the square of the antipode becomes the conjugation with the adjoined grouplike element. In Section 3, we study the character of the adjoint representation in order to prove that the character ring of a semisimple Hopf algebra is itself semisimple if the characteristic is sufficiently large. The results of both sections are combined in the final section to prove the stated result on Kaplansky's fifth conjecture.

All vector spaces are defined over a base field that is denoted by K. We assume familiarity with the basic notions of Hopf algebra theory that can be found for example in [13], [21], [30] or [33].