Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:56:17.506Z Has data issue: false hasContentIssue false
  • English
  • Français

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: donb@netspace.net.au
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: grovesd@maths.ox.ac.uk
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.

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

References

[1]Barnes, D. W., ‘On the cohomology of soluble Lie algebras’, Math. Z. 101 (1967), 343–349.CrossRefGoogle Scholar
[2]Barnes, D. W., ‘Sortability of representations of Lie algebras’, J. Algebra 27 (1973), 486–490.CrossRefGoogle Scholar
[3]Brandl, R., Franciosi, S. and de Giovanni, F., ‘On the Wielandt length of infinite soluble groups’, Glasgow Math. J. 32 (1990), 121–125.CrossRefGoogle Scholar
[4]Bryce, R. and Cossey, J., ‘The Wielandt subgroup of a finite group’, J. London Math. Soc. 40 (1989), 244–256.CrossRefGoogle Scholar
[5]Bryce, R., Cossey, J. and Ormerod, E., ‘A note on p-groups with power automorphisms’, Glasgow Math. J. 34 (1992), 327–332.CrossRefGoogle Scholar
[6]Camina, A., ‘The Wielandt length of finite groups’, J. Algebra 15 (1970), 142–148.CrossRefGoogle Scholar
[7]Casolo, C., ‘Soluble groups with finite Wielandt length’, Glasgow Math. J. 31 (1989), 329–334.CrossRefGoogle Scholar
[8]Casolo, C., ‘Wielandt series and defects of subnormal subgroups in finite soluble groups’, Rend. Sem. Mat. Univ. Padova 87 (1992), 93–104.Google Scholar
[9]Groves, D., The Wielandt ideal of a Lie algebra (M.Sc. Thesis, Australian National University, 1998).Google Scholar
[10]Hartley, B., ‘Locally nilpotent ideals of a Lie algebra’, Proc. Cambridge Philos. Soc. 63 (1967), 257–272.CrossRefGoogle Scholar
[11]Jacobson, N., Lie algebras (Interscience, New York, 1962).Google Scholar
[12]Schenkman, E., ‘A theory of subinvariant Lie algebras’, Amer. J. Math. 73 (1951), 453–474.CrossRefGoogle Scholar
[13]Stewart, I., Subideals of Lie algebras (Ph.D. Thesis, University of Warwick, 1969).Google Scholar
[14]Strade, H. and Farnsteiner, R., Modular Lie algebras and their representations, Monographs Textbooks Pure Appl. Math. 116 (Marcel Dekker, New York, 1988).Google Scholar
[15]Tuck, W., Frattini theory for Lie algebras (Ph.D. Thesis, University of Sydney, 1969).Google Scholar
[16]Wielandt, H., ‘Über den Normalisator der subnormalen Untergruppen’, Math. Z. 69 (1958), 463–465.CrossRefGoogle Scholar
[17]Zassenhaus, H., ‘On trace bilinear forms on Lie-algebras’, Proc. Glasgow Math. Assoc. 4 (1959), 62–72.CrossRefGoogle Scholar