×

Classifying pointed Hopf algebras of dimension 16. (English) Zbl 0952.16030

The authors classify the 43 pointed Hopf algebras of dimension 16 over an algebraically closed field of characteristic \(0\), up to isomorphism. It turns out that such a Hopf algebra is a lifting of a quantum linear space with Abelian coradical, or else is a group algebra. Each possible group of order less than or equal to 8 that could appear as the group of grouplikes is treated separately; the computations rely heavily on the Taft-Wilson theorem and the Nichols-Zoeller theorem.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andruskiewitsch N., J. Algebra
[2] Andruskiewitsch N. unpublished
[3] DOI: 10.1080/00927870008826844 · Zbl 0951.16014 · doi:10.1080/00927870008826844
[4] Beattie M., Constructing pointed Hopf algebras by Ore extensions · Zbl 0948.16026 · doi:10.1006/jabr.1999.8148
[5] Beattie M., Proc. Amer. Math. Soc
[6] Beattie M., Invent. Math
[7] DOI: 10.1006/jabr.1998.7504 · Zbl 0917.16016 · doi:10.1006/jabr.1998.7504
[8] Caenepeel S., Bull. London Math. Soc
[9] Chin W. unpublished
[10] DOI: 10.1006/jabr.1996.0145 · Zbl 0857.16036 · doi:10.1006/jabr.1996.0145
[11] Gelaki S., On Pointed Hopf algebras · Zbl 0857.16036
[12] Humphreys I.F., A Course in Group Theory (1996) · Zbl 0843.20001
[13] Kaplansky I., Bialgebras (1975) · Zbl 1311.16029
[14] Kassel C., Graduate Texts in Mathematics 155 (1995)
[15] Montgomery, S. Hopf algebras and their actions on rings. CBMS Reg. Conf. Series. Providence. Vol. 82, American Mathematical Society. · Zbl 0793.16029
[16] Montgomery, S. 1997. Classifying finite-dimensional semisimple Hopf algebras. Proceedings of the AMS-IMS-SIAM Summer Research Conference on Finite Dimensional Algebras. 1997, Seattle. AMS Contemp. Math. to appear
[17] DOI: 10.1080/00927877808822231 · Zbl 0405.16005 · doi:10.1080/00927877808822231
[18] DOI: 10.2307/2374514 · Zbl 0672.16006 · doi:10.2307/2374514
[19] DOI: 10.1090/S0002-9939-97-04143-9 · Zbl 0888.16022 · doi:10.1090/S0002-9939-97-04143-9
[20] Stefan D., Applications, J. Algebra 125 (1997)
[21] DOI: 10.1515/crll.1964.213.187 · Zbl 0125.01904 · doi:10.1515/crll.1964.213.187
[22] DOI: 10.1155/S1073792894000073 · Zbl 0822.16036 · doi:10.1155/S1073792894000073
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.