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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A characterization of the contact Lie algebras


Author: Thomas B. Gregory
Journal: Proc. Amer. Math. Soc. 82 (1981), 505-511
MSC: Primary 17B50; Secondary 17B20
MathSciNet review: 614868
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0614868-5
PII: S 0002-9939(1981)0614868-5
Article copyright: © Copyright 1981 American Mathematical Society