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Lattice automorphisms of semi-simple Lie algebras

Published online by Cambridge University Press:  09 April 2009

D. W. Barnes
D. W. Barnes
Affiliation:
Department of Pure Mathematics University of Sydney Australia
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Let L be a Lie algebra over the field F. A lattice automorphism of ℒ is an automorphism φ: ℒ(L) → ℒ(L) of the lattice ℒ(L) of all subalgebras of L. We seek to describe the lattice automorphisms in terms of maps σ: L→L of the underlying algebra. A semi-automorphism σ of L is an automorphism of the algebraic system consisting of the pair (F, L), that is, a pair of maps σ: F→ F, σ: L→ L preserving the operations. Thus σ: F → F is an automorphism of F and (x + y)σ = xσ + yσ, (xy)σ = xσyσ (λx)σ = λσxσ for all x, y ε L and λ ε F. Clearly, any semi-automorphism of L induces a lattice automorphism. To study a given lattice automorphism φ, we select a semi-automorphism a such that φσ-1 fixes certain subalgebras, and so we reduce the problem to the investigation of lattice automorphisms leaving these subalgebras fixed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 1 , August 1973 , pp. 43 - 53
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Barnes, D. W., ‘Lattice isomorphisms of Lie algebras’, J. Austral. Math. Soc. 4(1964), 470–475.CrossRefGoogle Scholar
[2]Barnes, D. W. and Wall, G. E., ‘On normaliser preserving lattice isomorphisms between nil-potent groups’, J. Austral. Math. Soc. 4 (1964), 454–469.CrossRefGoogle Scholar
[3]Jacobson, N., Lie Algebras (Interscience Tracts No. 10, 1962).Google Scholar