Hostname: page-component-6bf8c574d5-gr6zb Total loading time: 0 Render date: 2025-02-15T06:10:00.606Z Has data issue: false hasContentIssue false
  • English
  • Français

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: donwb@iprimus.com.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Barnes, D. W., ‘On -hypercentral modules for Lie algebras’, Arch. Math. 30 (1978), 1–7.CrossRefGoogle Scholar
[2]Barnes, D. W., ‘Saturated formations of soluble Lie algebras in characteristic 0’, Arch. Math. 30 (1978), 477–480.CrossRefGoogle Scholar
[3]Barnes, D. W., ‘Finitely sorting Lie algebras’, J. Austral. Math. Soc. 49 (1990), 354–363.CrossRefGoogle Scholar
[4]Barnes, D. W., ‘On -hyperexcentric modules for Lie algebras’, J. Austral. Math. Soc. 74 (2003), 235–238.CrossRefGoogle Scholar
[5]Barnes, D. W. and Gastineau-Hills, H. M., ‘On the theory of soluble Lie algebras’, Math. Z. 106 (1968), 343–354.CrossRefGoogle Scholar
[6]Harish-Chandra, , ‘Faithful representations of Lie algebras’, Ann. of Math. (2) 50 (1949), 68–76.CrossRefGoogle Scholar
[7]Hochschild, G., ‘An addition to Ado's Theorem’, Proc. Amer. Math. Soc. 17 (1966), 531–533.Google Scholar
[8]Iwasawa, K., ‘On the representation of Lie algebras’, Japan J. Math. 19 (1948), 405–426.Google Scholar
[9]Jacobson, N., Lie algebras (Interscience, New York, 1962).Google Scholar
[10]Schenkman, E., ‘A theory of subinvariant Lie algebras’, Amer. J. Math. 73 (1951), 453–474.CrossRefGoogle Scholar
[11]Strade, H. and Farnsteiner, R., Modular Lie algebras and their representations (Marcel Dekker, New York, 1988).Google Scholar