Cartan subalgebras of simple Lie algebras
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- by Robert Lee Wilson
- Trans. Amer. Math. Soc. 234 (1977), 435-446
- DOI: https://doi.org/10.1090/S0002-9947-1977-0480650-9
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Correction: Trans. Amer. Math. Soc. 305 (1988), 851-855.
Abstract:
Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic $p > 7$. Let H be a Cartan subalgebra of L, let $L = H + {\Sigma _{\gamma \in \Gamma }}{L_\gamma }$ be the Cartan decomposition of L with respect to H, and let $\bar H$ be the restricted subalgebra of Der L generated by ad H. Let T denote the maximal torus of $\bar H$ and I denote the nil radical of $\bar H$. Then $\bar H = T + I$. Consequently, each $\gamma \in \Gamma$ is a linear function on H.References
- John R. Schue, Cartan decompositions for Lie algebras of prime characteristic, J. Algebra 11 (1969), 25–52. MR 231873, DOI 10.1016/0021-8693(69)90099-4
- G. B. Seligman, Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, Springer-Verlag New York, Inc., New York, 1967. MR 0245627
- Helmut Strade, Nodale nichtkommutative Jordanalgebren und Lie-Algebren bei Charakteristik $p>2$, J. Algebra 21 (1972), 353–377 (German). MR 330246, DOI 10.1016/0021-8693(72)90001-4
- Robert Lee Wilson, The roots of a simple Lie algebra are linear, Bull. Amer. Math. Soc. 82 (1976), no. 4, 607–608. MR 409579, DOI 10.1090/S0002-9904-1976-14129-8
- David J. Winter, On the toral structure of Lie $p$-algebras, Acta Math. 123 (1969), 69–81. MR 251095, DOI 10.1007/BF02392385
Bibliographic Information
- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 234 (1977), 435-446
- MSC: Primary 17B20
- DOI: https://doi.org/10.1090/S0002-9947-1977-0480650-9
- MathSciNet review: 0480650