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 Izv. Akad. Nauk SSSR Ser. Mat., 1986, Volume 50, Issue 4, Pages 788–800 (Mi izv1535)

On Cartan subalgebras of Lie $p$-algebras

A. A. Premet

Abstract: It is shown, using the technique of switching toral subalgebras, that in finitedimensional Lie $p$-algebras every Cartan subalgebra with maximal toral part has dimension equal to the rank of the algebra. As is known, every Cartan subalgebra of a Lie $p$-algebra $\mathfrak g$ is of the form $\mathfrak g_x^0$, where $\mathfrak g_x^0$ is the nilspace of the endomorphism $\operatorname{ad}x$, $x\in\mathfrak g$. It is proved that there exists a Zariski-open subset $V\subset\mathfrak g$ such that for every $x\in V$ the subspace $\mathfrak g_x^0$ is a Cartan subalgebra with maximal toral part. A further result is the proof that the class of Cartan subalgebras with maximal toral part is the same as the class of Cartan subalgebras with minimal nilpotent part. The results are used to settle a question concerning anisotropic forms of Lie algebras over finite fields.
Bibliography: 12 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1987, 29:1, 145–157

Bibliographic databases:

UDC: 512.554
MSC: Primary 17B05; Secondary 17B50

Citation: A. A. Premet, “On Cartan subalgebras of Lie $p$-algebras”, Izv. Akad. Nauk SSSR Ser. Mat., 50:4 (1986), 788–800; Math. USSR-Izv., 29:1 (1987), 145–157

Citation in format AMSBIB
\Bibitem{Pre86} \by A.~A.~Premet \paper On Cartan subalgebras of Lie $p$-algebras \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1986 \vol 50 \issue 4 \pages 788--800 \mathnet{http://mi.mathnet.ru/izv1535} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=864177} \zmath{https://zbmath.org/?q=an:0633.17011|0613.17009} \transl \jour Math. USSR-Izv. \yr 1987 \vol 29 \issue 1 \pages 145--157 \crossref{https://doi.org/10.1070/IM1987v029n01ABEH000964} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Premet, “Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras”, Math. USSR-Sb., 66:2 (1990), 555–570
2. N. A. Koreshkov, “Cartan Subalgebras with Engel Decomposition”, Math. Notes, 72:4 (2002), 589–592
3. A. G. Gein, “Finite-dimensional simple Lie algebras with a subalgebra lattice of length 3”, Russian Math. (Iz. VUZ), 56:10 (2012), 62–65
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