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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Simple algebras of Weyl type, II


Author: Kaiming Zhao
Journal: Proc. Amer. Math. Soc. 130 (2002), 1323-1332
MSC (2000): Primary 16W10, 16W25, 17B20, 17B65, 17B05, 17B68
Published electronically: October 25, 2001
MathSciNet review: 1879953
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Abstract: Over a field $\mathbb{F}$ of any characteristic, for a commutative associative algebra $A$, and for a commutative subalgebra $D$ of $\operatorname{Der}_{\mathbb{F}}(A)$, the vector space $A[D]$ which consists of polynomials of elements in $D$ with coefficients in $A$ and which is regarded as operators on $A$ forms naturally an associative algebra. It is proved that, as an associative algebra, $A[D]$ is simple if and only if $A$ is $D$-simple. Suppose $A$ is $D$-simple. Then, (a) $A[D]$ is a free left $A$-module; (b) as a Lie algebra, the subquotient $[A[D],{A[D]}\,]/(Z(A[D])\cap [A[D], {A[D]}\,])$ is simple (except for one case), where $Z(A[D])$ is the center of $A[D]$. The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.


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Additional Information

Kaiming Zhao
Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Email: kzhao@math08.math.ac.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06218-9
PII: S 0002-9939(01)06218-9
Keywords: Simple Lie algebra, simple associative algebra, derivation
Received by editor(s): August 28, 2000
Received by editor(s) in revised form: November 20, 2000
Published electronically: October 25, 2001
Additional Notes: This work was supported by the Hundred Talents Program of Chinese Academy of Sciences and by NSF of China
Communicated by: Lance W. Small
Article copyright: © Copyright 2001 American Mathematical Society