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Linkable Dynkin diagrams. (English) Zbl 1062.16042

Summary: In this article we develop some aspects of the construction of new Hopf algebras found recently by N. Andruskiewitsch and H.-J. Schneider [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, No. 1, 1-26 (2002; Zbl 1007.16028)]. There the authors classified (under some slight restrictions) all pointed finite-dimensional Hopf algebras with coradical \((\mathbb{Z}/p)^s\). We contribute to this work by giving a closer description of the possible “exotic” linkings.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations

Citations:

Zbl 1007.16028
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References:

[1] Andruskiewitsch, N.; Schneider, H.-J., Finite quantum groups over abelian groups of prime exponent, Ann. Sci. École Norm. Super., 35, 1-26 (2002) · Zbl 1007.16028
[2] Andruskiewitsch, N.; Schneider, H.-J., Finite quantum groups and Cartan matrices, Adv. Math., 154, 1-45 (2000) · Zbl 1007.16027
[3] Andruskiewitsch, N.; Schneider, H.-J., Lifting of quantum linear spaces and pointed Hopf algebras of order \(p^3\), J. Algebra, 209, 659-691 (1998) · Zbl 0919.16027
[4] Andruskiewitsch, N.; Schneider, H.-J., Lifting of Nichols algebras of type \(A_2\) and pointed Hopf algebras of order \(p^4\), (Caenepeel, S.; Van Oystaeyen, F., Hopf Algebras and Quantum Groups. Hopf Algebras and Quantum Groups, Proceedings of the Brussels Conference. Hopf Algebras and Quantum Groups. Hopf Algebras and Quantum Groups, Proceedings of the Brussels Conference, Lecture Notes in Pure and Appl. Math., 209 (2000), Dekker: Dekker New York), 1-14 · Zbl 1020.16022
[5] N. Andruskiewitsch, H.-J. Schneider, A characterization of quantum groups, preprint, 2002, available at http://www.mathematik.uni-muenchen.de/ hanssch/Publications.html; N. Andruskiewitsch, H.-J. Schneider, A characterization of quantum groups, preprint, 2002, available at http://www.mathematik.uni-muenchen.de/ hanssch/Publications.html
[6] M. Beattie, S. Dăscălescu, S. Raianu, Lifting of Nichols algebras of type \(B_2\) http://www.mta.ca/ mbeattie/research/prepr.htm; M. Beattie, S. Dăscălescu, S. Raianu, Lifting of Nichols algebras of type \(B_2\) http://www.mta.ca/ mbeattie/research/prepr.htm
[7] Lusztig, G., Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc., 3, 257-296 (1990) · Zbl 0695.16006
[8] Kac, V., Infinite dimensional Lie Algebras (1995), Cambridge Univ. Press
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