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The Wielandt Subalgebra of a Lie Algebra

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

Donald W. Barnes  and
Daniel Groves
Donald W. Barnes
Affiliation:
Little Wonga Rd Cremorne NSW 2090 Australia e-mail: [email protected]
Daniel Groves*
Affiliation:
Department of Mathematics School of Advanced Studies Australian National University ACT 0200, Australia
*
Mathematical Institute 24–29 St. Giles Oxford, OX1 3LB UK e-mail: [email protected]
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Abstract

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Following the analogy with group theory, we define the Wielandt subalgebra of a finite-dimensional Lie algebra to be the intersection of the normalisers of the subnormal subalgebras. In a non-zero algebra, this is a non-zero ideal if the ground field has characteristic 0 or if the derived algebra is nilpotent, allowing the definition of the Wielandt series. For a Lie algebra with nilpotent derived algebra, we obtain a bound for the derived length in terms of the Wielandt length and show this bound to be best possible. We also characterise the Lie algebras with nilpotent derived algebra and Wielandt length 2.

Keywords

Lie algebrassubnormal subalgebras.

MSC classification

Secondary: 17B30: Solvable, nilpotent (super)algebras 17B50: Modular Lie (super)algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 3 , June 2003 , pp. 313 - 330
Copyright
Copyright © Australian Mathematical Society 2003

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