## 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