A characterization of the contact Lie algebras

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.