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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 Kharchenkoâ€™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

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.