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)
Full Text: DOI arXiv


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.