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A note on the conjugacy of Cartan subalgebras

Part of: Lie algebras and Lie superalgebras Linear algebraic groups and related topics

Published online by Cambridge University Press:  09 April 2009

David J. Winter
David J. Winter
Affiliation:
Department of MathematicsUniversity of MichiganAnn Arbor, Michigan 48104, U.S.A.
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Abstract

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The conjugacy of Cartan subalgebras of a Lie algebra L over an algebraically closed field under the connected automorphism group G of L is inherited by those G-stable ideals B for which B/Ci is restrictable for some hypercenter Ci of B. Concequently, if L is a restrictable Lie algebra such that L/Ci restrictable for some hypercenter Ci of L, and if the Lie algebra of Aut L contains ad L, then the Cartan subalgebras of L are conjugate under G. (The techniques here apply in particular to Lie algebras of characteristic 0 and classical Lie algebras, showing how the conjugacy of Cartan subgroups of algebraic groups leads quickly in these cases to the conjugacy of Cartan subalgebras.)

MSC classification

Secondary: 17B40: Automorphisms, derivations, other operators 17B45: Lie algebras of linear algebraic groups 20G15: Linear algebraic groups over arbitrary fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 2 , March 1979 , pp. 163 - 166
Copyright
Copyright © Australian Mathematical Society 1979

References

Chevalley, C. (1951), Théorie des groupes de Lie, Tome 2 (Hermann, Paris).Google Scholar
Humphreys, J. (1967), ‘Algebraic groups and modular Lie algebras’, Memoirs of the Amer. Math. Soc. 71, 1–76.Google Scholar
Jacobson, N. (1962), Lie algebras (Interscience, New York).Google Scholar
Seligman, G. (1957), Some remarks on classical Lie algebras, J. Math. Mech. 6, 549–558.Google Scholar
Seligman, G. (1978), Modular Lie algebras (Ergenbnisse 40, Springer-Verlag, New York).Google Scholar
Winter, D. J. (1970), ‘Cartan subalgebras of a Lie algebra and its ideals’, Pacific J. Math. 33 (2), 537–541.CrossRefGoogle Scholar
Winter, D. J. (1972), Abstract Lie algebras (Mass. Inst. Tech. Press).Google Scholar