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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

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