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On Noetherian affine prime regular Hopf algebras of Gelfand-Kirillov dimension 1

Author: Gongxiang Liu
Journal: Proc. Amer. Math. Soc. 137 (2009), 777-785
MSC (2000): Primary 16W30
Published electronically: October 29, 2008
MathSciNet review: 2457414
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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.

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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

Keywords: Homological integral, Gelfand-Kirillov dimension
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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