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Smash products and differential identities


Authors: Chen-Lian Chuang and Yuan-Tsung Tsai
Journal: Trans. Amer. Math. Soc. 364 (2012), 4155-4168
MSC (2010): Primary 16S40, 16S32, 16W25, 16S36, 16S30
DOI: https://doi.org/10.1090/S0002-9947-2012-05454-7
Published electronically: March 21, 2012
MathSciNet review: 2912449
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Abstract: Let $ \mathbf {U}$ be the universal enveloping algebra of a Lie algebra and $ R$ a $ \mathbf {U}$-module algebra, where $ \mathbf {U}$ is considered as a Hopf algebra canonically. We determine the centralizer of $ R$ in $ R\char93 \mathbf {U}$ with its associated graded algebra. We then apply this to the Ore extension $ R[X;\phi ]$, where $ \phi :X\to \mathrm {Der}(R)$. With the help of PBW-bases, the following is proved for a prime ring $ R$: Let $ Q$ be the symmetric Martindale quotient ring of $ R$. For $ f_i,g_i\in Q[X;\phi ]$, $ \sum _if_irg_i=0$ for all $ r\in R$ iff $ \sum _if_i\otimes g_i=0$, where $ \otimes $ is over the centralizer of $ R$ in $ Q[X;\phi ]$. Finally, we deduce from this Kharchenko's theorem on differential identities.


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Additional Information

Chen-Lian Chuang
Affiliation: Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
Email: chuang@math.ntu.edu.tw

Yuan-Tsung Tsai
Affiliation: Department of Applied Mathematics, Tatung University, Taipei 104, Taiwan
Email: yttsai@ttu.edu.tw

DOI: https://doi.org/10.1090/S0002-9947-2012-05454-7
Keywords: Derivations, universal enveloping algebras, centralizers, smash products, Ore extensions, differential identities
Received by editor(s): May 4, 2010
Received by editor(s) in revised form: August 30, 2010
Published electronically: March 21, 2012
Dedicated: To Pjek-Hwee Lee on his retirement
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.