A characterization of the contact Lie algebras
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- by Thomas B. Gregory PDF
- Proc. Amer. Math. Soc. 82 (1981), 505-511 Request permission
Abstract:
We classify the simple finite-dimensional irreducible graded Lie algebras over an algebraically closed field of characteristic $p > 5$ which have the form ${L_{ - 2}} \oplus {L_{ - 1}} \oplus {L_0} \oplus {L_1} \oplus \cdots \oplus {L_k},k \geqslant 3$, where ${L_0}$ is classical and reductive. We show that any such Lie algebra must be a Lie algebra of the contact series of Lie algebras of Cartan type by showing how the constraints imposed by the hypotheses force the existence of a highest-weight vector in ${L_{ - 1}}$ for the representation of ${L_0}$ in ${L_{ - 1}}$ induced by the adjoint representation of $L$ in itself. The existence of this highest-weight vector enables us to conclude that the above-mentioned representation is restricted. $L$ can then be determined by appeal to an earlier classification theorem.References
- Thomas B. Gregory, Simple Lie algebras with classical reductive null component, J. Algebra 63 (1980), no. 2, 484–493. MR 570725, DOI 10.1016/0021-8693(80)90085-X
- V. G. Kac, The classification of the simple Lie algebras over a field with non-zero characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 385–408 (Russian). MR 0276286
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 505-511
- MSC: Primary 17B50; Secondary 17B20
- DOI: https://doi.org/10.1090/S0002-9939-1981-0614868-5
- MathSciNet review: 614868