Cartan Subalgebras of Zassenhaus Algebras

Gordon Brown

doi:10.4153/CJM-1975-105-4

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Cartan subalgebras play a very important role in the classification of the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic zero. It is well-known [5, 273] that any two Cartan subalgebras of such an algebra are conjugate, i.e. images of one another under some automorphism of the algebra. On the other hand, there exist finitedimensional simple Lie algebras over fields of finite characteristic p possessing non-conjugate Cartan subalgebras [2; 3; 4]. The simple Lie algebras discovered by Zassenhaus [6] also possess non-conjugate Cartan subalgebras, and we shall give a complete classification of Cartan subalgebras of these algebras in this paper.

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3. Demuskin, S. P., Cartan subalgebras of the simple Lie p-algebras Wn and Sn, Sibirsk. Mat. Z 11 (1970), 310–325.Google Scholar

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5. Jacobson, Nathan, Lie Algebras (Interscience, New York, 1962).Google Scholar

6. Zassenhaus, Hans, Uber Liesche Ringe mit Primzahlcharakteristik, Hamb. Ahb. 13 (1939), 1–100.Google Scholar

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