On graded Lie algebras of characteristic three with classical reductive null component

Abstract

We consider finite-dimensional irreducible transitive graded Lie algebras L = r i=−q Li over algebraically closed fields of characteristic three. We assume that the null component L0 is classical and reductive. The adjoint representation of L on itself induces a representation of the commutator subalgebra L ′ 0 of the null component on the minus-one component L−1. We show that if the depth q of L is greater than one, then this representation must be restricted.Over algebraically closed fields F of characteristic p > 0, the classification of the finite-dimensional simple Lie algebras relies on the classification of the finite-dimensional irreducible transitive graded Lie algebras L = r i=−q L i of depth q ≧ 1 with classical reductive null component L 0 . We recall some of the progress that has been made in the classification of such Lie algebras L. In the case in which L −1 is not only irreducible but also restricted as an L 0 -module, such

On graded Lie algebras of characteristic three with classical reductive null component

“…We have noted that the Main Theorem has been proved for q = 2 in [2] and for q = 3 in [7]. When we refer to the Main Theorem to substantiate certain claims below, it will be for the cases already proved.…”

“…When we make use of certain properties of these Lie algebras in later sections, we will explicitly state the properties we need. Proposition 1.6 (See Lemma 2.12 of [2].) Let L = L −1 ⊕L 0 ⊕L 1 ⊕· · ·⊕L r be a graded Lie algebra satisfying conditions (A), (B), and (C) of the Main Theorem, and suppose that L 1 = 0.…”

“…If L −1 is a nonrestricted L ′ 0 −module, then either L is isomorphic to one of the Lie algebras L(ǫ) or M , or L is a Hamiltonian Lie algebra such that H(2 : n, ω) ⊆ L ⊆ CH(2 : n, ω), where n = (1, n 2 ), ω = (exp x (3) )dx ∧ dy, and the grading is of type (0,1). [2].) Let L = L −1 ⊕ L 0 ⊕ L 1 ⊕ · · · ⊕ L q be one of the Lie algebras L(ǫ), M, or H(2 : n, ω) with n = (1, n 2 ), let χ be the nonzero character of the L 0 -module L −1 , and let V be an L-module such that…”

“…For q = 3, the corresponding statement was proved in [7]. It was conjectured in [2] that a non-restricted L ′ 0 −module L −1 can exist only in Lie algebras of depth one. The present paper completes the proof of that conjecture for r > 1.…”

“…We require that r be greater than one in order to exclude, for example, H(2 : n, ω) with the reverse gradation; see also Example 0.3 below. We note, as in [2], that because there are only finitely many irreducible restricted modules for the derived algebra of a classical reductive Lie algebra, what needs to be considered in classifying graded Lie algebras over algebraically closed fields of characteristic three is reduced. In this paper, we prove the following theorem, which we will henceforth refer to as the "Main Theorem."…”