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 p > 5 which have the form L − 2 ⊕ L − 1 ⊕ L 0 ⊕ L 1 ⊕ ⋯ ⊕ L k , k ⩾ 3 {L_{ - 2}} \oplus {L_{ - 1}} \oplus {L_0} \oplus {L_1} \oplus \cdots \oplus {L_k},k \geqslant 3 , where L 0 {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 {L_{ - 1}} for the representation of L 0 {L_0} in L − 1 {L_{ - 1}} induced by the adjoint representation of L L in itself. The existence of this highest-weight vector enables us to conclude that the above-mentioned representation is restricted. L L can then be determined by appeal to an earlier classification theorem.