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Commuting rings of simple A(k)-modules

Published online by Cambridge University Press:  09 April 2009

Daniel R. Farkas  and
Robert L. Snider
Daniel R. Farkas
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
Robert L. Snider
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
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Abstract

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For the Weyl algebra A(k) and each finite dimensional division ring D over k, there exists a simple A(k)-module whose commuting ring is D.

It has been known for some time that if A(k) denotes the Weyl algebra over a field k of characteristic zero, the commuting ring of a simple A(k)-module is a division algebra finite dimensional over k (see the introduction of [1]). Which division algebras actually appear? Quebbemann [1] showed that if D is a finite dimensional division algebra whose center is k, then it occurs as a commuting ring. We complete this circle of ideas by showing that any D appears: a division algebra over k appears as the commuting ring of a simple A(k)-module if and only if it is finite dimensional over k.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 2 , August 1981 , pp. 142 - 145
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Quebbemann, H. G., ‘Schiefkörper als Endomorphismenringe einfacher Moduln über einer Weyl-Algebra’, J. of Alg. 59 (1979), 311–312.CrossRefGoogle Scholar