Existence of ad-nilpotent elements and simple Lie algebras with subalgebras of codimension one
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Abstract:
For a perfect field $F$ of arbitrary characteristic, the following statements are proved to be equivalent: (1) Any Lie algebra over $F$ contains an ad-nilpotent element. (2) There are no simple Lie algebras over $F$ having only abelian subalgebras. They are used to guarantee the existence of an ad-nilpotent element in every Lie algebra over a perfect field of type $({C_1})$ of arbitrary characteristic (in particular, over any finite field). Furthermore, we give a sufficient condition to insure the existence of ad-nilpotent elements in a Lie algebra over any perfect field. As a consequence of this result we obtain an easy proof of the fact that the Zassenhaus algebras and ${\text {sl}}(2,F)$ are the only simple Lie algebras which have subalgebras of codimension 1, whenever the ground field $F$ is perfect with ${\text {char}}(F) \ne 2$. All Lie algebras considered are finite dimensional.References
- Ralph K. Amayo, Quasi-ideals of Lie algebras. I, Proc. London Math. Soc. (3) 33 (1976), no. 1, 28–36. MR 409573, DOI 10.1112/plms/s3-33.1.28
- Donald W. Barnes, On Cartan subalgebras of Lie algebras, Math. Z. 101 (1967), 350–355. MR 220785, DOI 10.1007/BF01109800
- G. M. Benkart and I. M. Isaacs, On the existence of ad-nilpotent elements, Proc. Amer. Math. Soc. 63 (1977), no. 1, 39–40. MR 432721, DOI 10.1090/S0002-9939-1977-0432721-6
- G. M. Benkart, I. M. Isaacs, and J. M. Osborn, Lie algebras with self-centralizing ad-nilpotent elements, J. Algebra 57 (1979), no. 2, 279–309. MR 533800, DOI 10.1016/0021-8693(79)90225-4
- A. S. Dzhumadil′daev, Simple Lie algebras with a subalgebra of codimension one, Uspekhi Mat. Nauk 40 (1985), no. 1(241), 193–194 (Russian). MR 783616
- Rolf Farnsteiner, On $\textrm {ad}$-semisimple Lie algebras, J. Algebra 83 (1983), no. 2, 510–519. MR 714262, DOI 10.1016/0021-8693(83)90236-3
- Helmut Strade and Rolf Farnsteiner, Modular Lie algebras and their representations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker, Inc., New York, 1988. MR 929682
- Karl Heinrich Hofmann, Lie algebras with subalgebras of co-dimension one, Illinois J. Math. 9 (1965), 636–643. MR 181704
- Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
- Masayoshi Nagata, Field theory, Pure and Applied Mathematics, No. 40, Marcel Dekker, Inc., New York-Basel, 1977. MR 0469887
- A. A. Premet, Toroidal Cartan subalgebras of Lie $p$-algebras, and anisotropic Lie algebras of positive characteristic, Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 1 (1986), 9–14, 124 (Russian, with English summary). MR 837626
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 363-368
- MSC: Primary 17B40; Secondary 17B50
- DOI: https://doi.org/10.1090/S0002-9939-1988-0962799-X
- MathSciNet review: 962799