## Smash products and differential identities

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- by Chen-Lian Chuang and Yuan-Tsung Tsai PDF
- Trans. Amer. Math. Soc.
**364**(2012), 4155-4168 Request permission

## 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 $R\#\mathbf {U}$ 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.## References

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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
- Received by editor(s): May 4, 2010
- Received by editor(s) in revised form: August 30, 2010
- Published electronically: March 21, 2012
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2912449

Dedicated: To Pjek-Hwee Lee on his retirement