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 Mat. Sb., 1989, Volume 180, Number 4, Pages 542–557 (Mi msb2991)

Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras

A. A. Premet

Abstract: Let $\mathfrak G$ be a finite-dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$. It is proved that any two Cartan subalgebras with maximal toral part in $\mathfrak G$ can be obtained from each other by means of a finite chain of elementary transformations that are similar in form to the exponents of the inner root derivations of $\mathfrak G$. The following theorem plays an important role in the proof:
Theorem. {\it Let $s$ be a toral rank of $\mathfrak G$ and $e_1,…,e_n$ a basis of $\mathfrak G$. There exists $\nu\in\mathbf Z_+$ and homogeneous polynomials $f_0,…,f_{s-1},$ in $n$ variables$,$ such that
$$x^{[p^{s+\nu}]}=\sum_{i=0}^{s-1}f_i(x_1,…,x_n)x^{[p^{i+\nu}]}$$
$($here $x=x_1e_1+…+x_ne_n$ and $\deg f_i=p^{s+\nu}-p^{i+\nu}).$}
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Sbornik, 1990, 66:2, 555–570

Bibliographic databases:

UDC: 512.554
MSC: Primary 17B05, 17B20; Secondary 17B40, 17B30

Citation: A. A. Premet, “Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras”, Mat. Sb., 180:4 (1989), 542–557; Math. USSR-Sb., 66:2 (1990), 555–570

Citation in format AMSBIB
\Bibitem{Pre89} \by A.~A.~Premet \paper Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras \jour Mat. Sb. \yr 1989 \vol 180 \issue 4 \pages 542--557 \mathnet{http://mi.mathnet.ru/msb2991} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=997900} \zmath{https://zbmath.org/?q=an:0691.17007|0698.17008} \transl \jour Math. USSR-Sb. \yr 1990 \vol 66 \issue 2 \pages 555--570 \crossref{https://doi.org/10.1070/SM1990v066n02ABEH002084} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1990DY49300016} 

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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. S. A. Tyurin, “The $p$-operation in the Zassenhaus algebra”, Russian Math. (Iz. VUZ), 42:1 (1998), 78–80
2. Skryabin S., “Toral Rank One Simple Lie Algebras of Low Characteristics”, J. Algebra, 200:2 (1998), 650–700
3. S. A. Tyurin, “Surgeries of tori in the Zassenhaus algebra”, Russian Math. (Iz. VUZ), 43:4 (1999), 53–59
4. Premet A. Strade H., “Simple Lie Algebras of Small Characteristic II. Exceptional Roots”, J. Algebra, 216:1 (1999), 190–301
5. Skryabin, S, “Tori in the Melikian algebra”, Journal of Algebra, 243:1 (2001), 69
6. Premet A. Strade H., “Simple Lie Algebras of Small Characteristic III. the Toral Rank 2 Case”, J. Algebra, 242:1 (2001), 236–337
7. N. A. Koreshkov, “Cartan Subalgebras with Engel Decomposition”, Math. Notes, 72:4 (2002), 589–592
8. Skryabin S., “Invariant Polynomial Functions on the Poisson Algebra in Characteristic P”, J. Algebra, 256:1 (2002), 146–179
9. Premet A. Strade H., “Simple Lie Algebras of Small Characteristic IV. Solvable and Classical Roots”, J. Algebra, 278:2 (2004), 766–833
10. Premet A. Strade H., “Classification of Finite Dimensional Simple Lie Algebras in Prime Characteristics”, Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, Contemporary Mathematics, 413, ed. Benkart G. Jantzen J. Lin Z. Nakano D. Parshall B., Amer Mathematical Soc, 2006, 185–214
11. Amiram Braun, Gil Vernik, “On the center and semi-center of enveloping algebras in prime characteristic”, Journal of Algebra, 322:5 (2009), 1830
12. Hao Chang, Yu-Feng Yao, “On
$${\mathbb{F}_q}$$
F q -Rational Structure of Nilpotent Orbits in the Witt Algebra”, Results. Math, 2013
13. Yu.F.eng Yao, Hao Chang, “Borel subalgebras of the Witt algebra”, Acta. Math. Sin.-English Ser, 31:8 (2015), 1348
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