On Noetherian affine prime regular Hopf algebras of Gelfand-Kirillov dimension 1
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Abstract:
Let $k$ be an algebraically closed field. In 2007, D.-M. Lu, Q.-S. Wu, and J. J. Zhang asked the following question: Besides the group algebras $k\mathbb {Z},\;k\mathbb {D}$ and infinite dimensional prime Taft algebras, are there other noetherian affine prime regular Hopf algebras of GK-dimension 1? In this paper, we give a new one. Another problem posed by Lu, Wu, and Zhang can also be resolved by this example. Assuming $H$ is a noetherian affine prime regular Hopf algebra of GK-dimension 1, we show that gr$H:= \bigoplus _{s\geq 0}J_{iq}^{s}/J_{iq}^{s+1}$, as a Hopf algebra, is isomorphic to an infinite dimensional prime Taft algebra. This gives a characterization of infinite dimensional prime Taft algebras.References
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Additional Information
- Gongxiang Liu
- Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
- Address at time of publication: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
- MR Author ID: 766485
- Email: gxliu@nju.edu.cn
- Received by editor(s): September 25, 2006
- Received by editor(s) in revised form: March 25, 2007
- Published electronically: October 29, 2008
- Additional Notes: Project supported by the Natural Science Foundation of China (No. 10801069).
- Communicated by: Martin Lorenz
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 777-785
- MSC (2000): Primary 16W30
- DOI: https://doi.org/10.1090/S0002-9939-08-09034-5
- MathSciNet review: 2457414