Abstract
In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf subalgebra. For a non-cosemisimple Hopf algebra A with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When A is of finite dimension, we provide a necessary and sufficient condition for A’s diagram equaling the diagram of its maximal pointed Hopf subalgebra.
Similar content being viewed by others
References
N Andruskiewitsch, M Grana. Examples of liftings of Nichols algebras over racks, AMA Algebra Montp Announc, 2003, 2003(1): 6.
N Andruskiewitsch, I Heckenberger, HSchneider. The Nichols algebra of a semisimple Yetter-Drinfeld module, Amer J Math, 2010, 132(6): 1493–1547.
N Andruskiewitsch, S Natale. Counting arguments for Hopf algebras of low dimension, Tsukuba J Math, 2001, 25(1): 187–201.
N Andruskiewitsch, H Schneider. Lifting of quantum linear spaces and pointed Hopf algebras of order p3, J Algebra, 1998, 209(2): 658–691.
N Andruskiewitsch, H Schneider. Finite quantum groups and Cartan matrices, Adv Math, 2000, 154(1): 1–45.
N Andruskiewitsch, H Schneider. Pointed Hopf algebras, New directions in Hopf algebras, 2002, 43: 1–68.
N Andruskiewitsch, H Schneider. On the classification of finite-dimensional pointed Hopf algebras, Ann of Math, 2010, 171(1): 375–417.
N Andruskiewitsch, C Vay. Finite dimensional Hopf algebras over the dual group algebra of the symmetric group in three letters, Amer J Math, 2010, 132(6): 1493–1547.
B Bakalov, AD’ Andrea, V GKac. Theory of finite pseudoalgebras, Adv Math, 2001, 162(1): 1–140.
M Beattie, S Dăscălescu. Hopf algebras of dimension 14, J Lond Math Soc, 2004, 69(1): 65–78.
M Beattie, GA Garcia. Classifying Hopf algebras of a given dimension, Hopf Algebras and Tensor Categories, Contemp Math, 2013, 585: 125–152.
M Beattie, GA Garcia. Techniques for classifying Hopf algebras and applications to dimension p3, Comm Algebra, 2013, 41(8): 3108–3129.
C Călinescu, S Dăscălescu, AMasuoka, CMenini. Quantum lines over non-cocommutative cosemisimple Hopf algebras, J Algebra, 2004, 273(2): 753–779.
Z P Fan, DM Lu. Block Systems of finite dimensional non-cosemisimple Hopf Algebras, arxiv: 1509.05837.
D Fukuda. Structure of coradical filtration and its application to Hopf algebras of dimension pq, Glasg Math J, 2008, 50(2): 183–190.
I Kaplansky. Bialgebras, University of Chicago, Department of Mathematics, 1975.
A Masuoka. Cocycle deformations and Galois objects for some cosemisimple Hopf algebras of finite dimension, Contemp Math, 2000, 267: 195–214.
S Montgomery. Hopf algebras and their actions on rings, Amer Math Soc, 1993.
SH Ng. On the projectivity of module coalgebras, Proc Amer Math Soc, 1998, 126(11): 3191–3198.
DE Radford. The structure of Hopf algebras with a projection, J Algebra, 1985, 92(2): 322–347.
ME Sweedler. Hopf algebras, W A Benjamin, New York, 1969.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (11271319, 11301126).
Rights and permissions
About this article
Cite this article
Fan, Zp., Lu, Dm. & Yu, Xl. Diagrams of Hopf algebras with the Chevalley property. Appl. Math. J. Chin. Univ. 31, 367–378 (2016). https://doi.org/10.1007/s11766-016-3403-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-016-3403-2