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Ado-Iwasawa extras

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

Donald W. Barnes
Donald W. Barnes
Affiliation:
1 Little Wonga Road Cremorne NSW 2090Australia e-mail: [email protected]
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Abstract

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Let L be a finite-dimensional Lie algebra over the field F. The Ado-Iwasawa Theorem asserts the existence of a finite-dimensional L-module which gives a faithful representation ρ of L. Let S be a subnormal subalgebra of L, let be a saturated formation of soluble Lie algebras and suppose that S ∈ . I show that there exists a module V with the extra property that it is -hypercentral as S-module. Further, there exists a module V which has this extra property simultaneously for every such S and , along with the Hochschild extra that ρ(x) is nilpotent for every x ∈ L with ad(x) nilpotent. In particular, if L is supersoluble, then it has a faithful representation by upper triangular matrices.

Keywords

Lie algebrasfaithful representationssaturated formations

MSC classification

Secondary: 17B30: Solvable, nilpotent (super)algebras 17B56: Cohomology of Lie (super)algebras 17B50: Modular Lie (super)algebras 17B55: Homological methods in Lie (super)algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 3 , June 2005 , pp. 407 - 421
Copyright
Copyright © Australian Mathematical Society 2005

References

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