• Preprint: Series "Graduiertenkolleg Mathematik im Bereich ihrer Wechselwirkung mit der Physik": gk-mp-9806/55 (dvi, ps)
• Book: F. v. Oystaeyen/M. Saorin (ed.): Interactions between Ring Theory and Representations of Algebras, Lect. Notes Pure Appl. Math., Vol. 210, Dekker, New York, 2000, 393-412

We survey the known results on Kaplansky's ten conjectures on Hopf algebras.

In the autumn of 1973, I. Kaplansky gave a course on bialgebras in Chicago. For this course, he prepared some lecture notes that he originally intended to turn into a comprehensive account on the subject. In 1975, he changed his mind and published these lecture notes without larger additions. These lecture notes contain, besides a fairly comprehensive bibliography of the literature available at that time, two appendices. The first of these appendices is concerned with bialgebras of low dimension, whereas the second one contains a list of ten conjectures on Hopf algebras which are known today as Kaplansky's conjectures.

Kaplansky's conjectures did not arise as the product of a long investigation in the field of Hopf algebras; also, Kaplansky did not make many contributions to the solution of his conjectures. He only intended to list a number of interesting problems at the end of his lecture notes - lecture notes that he himself called informal. Because of this, it happened that one conjecture in the list was already solved at the time of publication, another one is very simple. That the conjectures nevertheless gained considerable importance for the field is due to the fact that Kaplansky achieved to touch upon a number of questions of fundamental character.

This article tries to summarize the present knowledge about Kaplansky's conjectures. Brief surveys can be found in [53] and [65], some conjectures are also discussed in [38] and [43]. Here, the exposition shall be more detailed, but nevertheless not comprehensive. Since Kaplansky's lecture notes are not always easily accessible, we have reproduced the conjectures in their original formulation in an appendix. The reader should note that, except for the appendix, the formulation of results does not follow literally the quoted sources. In addition, usually not all important results of a quoted article are mentioned. We also note that Kaplansky posed various other conjectures concerning different fields of mathematics; these are not discussed here.

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