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Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple. (English) Zbl 0649.16005

Let A be a finite-dimensional Hopf algebra over a field k. Kaplansky conjectured that if A is cosemisimple, then its antipode S has square \(S^ 2=I\) (i.e. S is an involution). The authors show that if A is cosemisimple and k has characteristic zero, then A is semisimple, from which it follows that \(S^ 4=I\). In a subsequent paper which was published before the one under review [Am. J. Math. 110, No.1, 187-195 (1988; Zbl 0637.16006)], the authors sharpened this to \(S^ 2=I\), thus verifying Kaplansky’s conjecture when k has characteristic zero.
The main tool of this paper is to associate an element \(\Lambda_ f\) of A (a finite-dimensional Hopf algebra) to each linear endomorphism f of A via \(p(\Lambda_ f)= \text{Trace}(L(p)\circ f^*)\) for all p in the dual (Hopf) algebra \(A^*\), where L is left multiplication in \(A^*\). Set \(x=\Lambda_ I\) and \({\tilde \Lambda}=\Lambda_{S^ 2}\). Then \({\tilde\Lambda}\) is a left integral of A (i.e., \(a{\tilde \Lambda}= \epsilon(a){\tilde\Lambda}\) for a in A, \(\epsilon\) the counit of A), and \({\tilde \Lambda}\neq 0\) iff A is cosemisimple. This is used to prove that A cosemisimple implies A semisimple at characteristic zero.
The authors also study the relation between \({\tilde\Lambda}\) and x carefully. If \((\dim A)1\neq 0\), then \(S^ 2=I\) iff x is a non-zero left integral of A (and then \(x={\tilde\Lambda}\) and A is both semisimple and cosemisimple). x is studied relative to the structure of A. Finally, when A is cosemisimple over an algebraically closed field of characteristic 0, the authors compute x and \({\tilde\Lambda}\) with respect to a basis of matrix counits of A.
Reviewer: E.J.Taft

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)

Citations:

Zbl 0637.16006
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References:

[1] Kaplansky, I., Bialgebras, (Lecture Notes in Mathematics (1975), University of Chicago: University of Chicago Chicago) · Zbl 1311.16029
[2] Larson, R. G., Characters of Hopf algebras, J. Algebra, 17, 352-368 (1971) · Zbl 0217.33801
[3] Larson, R. G.; Sweedler, M. E., An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math., 91, 75-94 (1969) · Zbl 0179.05803
[4] Radford, D. E., On the antipode of a cosemisimple Hopf algebra, J. Algebra, 1, 68-88 (1984) · Zbl 0531.16005
[5] Radford, D. E., The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math., 98, 333-355 (1976) · Zbl 0332.16007
[6] Sweedler, M. E., Hopf Algebras (1969), Benjamin: Benjamin New York · Zbl 0194.32901
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