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Irreducible Modules for Polycyclic Group Algebras

Published online by Cambridge University Press:  20 November 2018

I. M. Musson
I. M. Musson*
Affiliation:
The University of Wisconsin-Madison, Madison, Wisconsin
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If G is a polycyclic group and k an absolute field then every irreducible kG-module is finite dimensional [10], while if k is nonabsolute every irreducible module is finite dimensional if and only if G is abelian-by-finite [3]. However something more can be said about the infinite dimensional irreducible modules. For example P. Hall showed that if G is a finitely generated nilpotent group and V an irreducible kG-module, then the image of kZ in EndkGV is algebraic over k [3]. Here Z = Z(G) denotes the centre of G. It follows that the restriction Vz of V to Z is generated by finite dimensional kZ-modules. In this paper we prove a generalization of this result to polycyclic group algebras.

We introduce some terminology.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 4 , 01 August 1981 , pp. 901 - 914
Copyright
Copyright © Canadian Mathematical Society 1981

References

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