Lifting of quantum linear spaces and pointed Hopf algebras of order \(p^3\). (English) Zbl 0919.16027

Let \(k\) be an algebraically closed field of characteristic zero. Let \(A\) be a noncosemisimple Hopf algebra over \(k\) whose coradical \(A_0\) is a Hopf subalgebra, e.g., if \(A\) is pointed. Let \(\text{gr }A\) be the graded Hopf algebra associated to the coradical filtration of \(A\), \(\pi\) the projection of \(\text{gr }A\) on \(A_0\). Then \(R=\{a\) in \(\text{gr }A:(I\otimes\pi)\Delta a=a\otimes 1\}\), the coinvariants of \(\pi\), is a Hopf algebra in the braided category of Yetter-Drinfeld modules over \(A_0\). \(\text{gr }A\) can be recovered from \(R\) by bosonization (biproduct with \(A_0\)). The authors propose to study \(A\) by first studying \(R\), transferring the information to \(\text{gr }A\), and then lifting to \(A\). In this article, they apply this plan when \(R\) is a quantum linear space. This enables them to classify all pointed Hopf algebras of order \(p^3\), \(p\) an odd prime, and also to classify all pointed Hopf algebras whose coradical is Abelian and has index \(p\) or \(p^2\). Their lifting results then enable them to produce an infinite family of pointed, nonisomorphic Hopf algebras of dimension \(p^4\). This answers in the negative a conjecture of Kaplansky.
Reviewer’s remark: This conjecture has also been answered recently by other authors using different constructions.


16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W50 Graded rings and modules (associative rings and algebras)
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[1] Andruskiewitsch, N., Notes on extensions of Hopf algebras, Canad. J. Math., 48, 3-42 (1996) · Zbl 0857.16033
[2] Andruskiewitsch, N.; Schneider, H.-J, Simplicity of some pointed Hopf algebras, Appendix, Canad. J. Math., 48, 3-42 (1996)
[3] Andruskiewitsch, N.; Schneider, H.-J, Hopf Algebras of order \(p^2p\), J. Algebra, 199, 430-454 (1998) · Zbl 0899.16019
[5] Bergman, G., The diamond lemma for ring theory, Adv. Math., 29, 178-218 (1978) · Zbl 0326.16019
[7] Bourbaki, N., Algébre commutative. Charitre III (1961), Hermann
[9] Chin, W.; Musson, I., The coradical filtration for quantized universal enveloping algebras, J. London Math. Soc., 53, 50-67 (1996) · Zbl 0860.17013
[11] Kaplansky, I., Bialgebras (1975), University of Chicago: University of Chicago Chicago
[12] Larson, R. G.; Radford, D. E., Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra, 117, 267-289 (1988) · Zbl 0649.16005
[13] Lusztig, G., Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc., 3, 257-296 (1990) · Zbl 0695.16006
[14] Lusztig, G., Quantum groups at roots of 1, Geom. Dedicata, 35, 89-114 (1990) · Zbl 0714.17013
[15] Majid, S., Crossed products by braided groups and bosonization, J. Algebra, 163, 165-190 (1994) · Zbl 0807.16036
[16] Masuoka, A., Self dual Hopf algebras of dimension \(p^3\), J. Algebra, 178, 791-806 (1995) · Zbl 0840.16031
[17] Masuoka, A., The\(p^n\)theorem for semisimple Hopf algebras, Proc. Amer. Math. Soc., 124, 735-737 (1996) · Zbl 0848.16033
[18] Montgomery, S., Hopf Algebras and Their Actions on Rings (1993), Am. Math. Soc: Am. Math. Soc Providence · Zbl 0804.16041
[19] Nichols, W. D., Bialgebras of type one, Commun. Alg., 6, 1521-1552 (1978) · Zbl 0408.16007
[20] Nichols, W. D.; Zoeller, M. B., A Hopf algebra freeness theorem, Amer. J. Math., 111, 381-385 (1989) · Zbl 0672.16006
[21] Oberst, U.; Schneider, H.-J, Über Untergruppen endlicher algebraischer Gruppen, Manuscripta Math., 8, 217-241 (1973) · Zbl 0259.20043
[22] Radford, D., On the coradical of a finite dimensional Hopf algebra, Proc. Amer. Math. Soc., 53, 9-15 (1975) · Zbl 0324.16009
[23] Radford, D., The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math., 98, 333-355 (1976) · Zbl 0332.16007
[24] Radford, D., Hopf algebras with projection, J. Algebra, 92, 322-347 (1985) · Zbl 0549.16003
[26] Stefan, D., The set of types of \(n\), J. Algebra, 193, 571-580 (1997) · Zbl 0882.16029
[28] Sweedler, M., Hopf Algebras (1969), Benjamin: Benjamin New York
[29] Taft, E.; Wilson, R. L., On antipodes in pointed Hopf algebras, J. Algebra, 29, 27-32 (1974) · Zbl 0282.16008
[30] Takeuchi, M., Some topics on\(GL_q}(n\), J. Algebra, 147, 379-410 (1992) · Zbl 0760.16015
[31] Zhu, Y., Hopf algebras of prime dimension, Int. Math. Res. Notes, 1, 53-59 (1994) · Zbl 0822.16036
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