Simple Lie algebras of toral rank one

Wilson, Robert Lee

doi:10.1090/S0002-9947-1978-0463252-0

Simple Lie algebras of toral rank one
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by Robert Lee Wilson

Trans. Amer. Math. Soc. 236 (1978), 287-295

DOI: https://doi.org/10.1090/S0002-9947-1978-0463252-0

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Abstract:

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic $p > 7$. Let L have Cartan decomposition $L = H + {\sum _{\gamma \in \Gamma }}{L_\gamma }$. If $\Gamma$ generates a cyclic group then L is isomorphic to ${\text {sl}}(2,F)$ or to one of the simple Lie algebras of generalized Cartan type $W(1:{\mathbf {n}})$ or $H{(2:{\mathbf {n}}:\Phi )^{(2)}}$.

References

Richard E. Block, On Lie algebras of rank one, Trans. Amer. Math. Soc. 112 (1964), 19–31. MR 160803, DOI 10.1090/S0002-9947-1964-0160803-0
Gordon Brown, Cartan subalgebras of Zassenhaus algebras, Canadian J. Math. 27 (1975), no. 5, 1011–1021. MR 401851, DOI 10.4153/CJM-1975-105-4
Ho-Jui Chang, Über Wittsche Lie-Ringe, Abh. Math. Sem. Hansischen Univ. 14 (1941), 151–184 (German). MR 5100, DOI 10.1007/BF02940743
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842, DOI 10.1007/978-1-4612-6398-2
N. Jacobson, A note on three dimensional simple Lie algebras, J. Math. Mech. 7 (1958), 823–831. MR 0097432, DOI 10.1512/iumj.1958.7.57047
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

Description of filtered Lie algebras with which graded Lie algebras of Cartan type are associated

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Irving Kaplansky, Lie algebras of characteristic $p$, Trans. Amer. Math. Soc. 89 (1958), 149–183. MR 99359, DOI 10.1090/S0002-9947-1958-0099359-7
A. I. Kostrikin and I. R. Šafarevič, Graded Lie algebras of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 251–322 (Russian). MR 0252460
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
Robert Lee Wilson, A structural characterization of the simple Lie algebras of generalized Cartan type over fields of prime characteristic, J. Algebra 40 (1976), no. 2, 418–465. MR 412239, DOI 10.1016/0021-8693(76)90206-4
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
Robert Lee Wilson, Cartan subalgebras of simple Lie algebras, Trans. Amer. Math. Soc. 234 (1977), no. 2, 435–446. MR 480650, DOI 10.1090/S0002-9947-1977-0480650-9