Graded Lie Algebras
In this chapter, we develop results about general graded Lie algebras.
Later (starting in Section 1.8) and in subsequent chapters, we specialize to
modular graded Lie algebras satisfying the hypotheses of the Recognition
To begin, our focus is on Lie algebras over an arbitrary field F having
an integer grading,
where [gi, gj ] ⊆ gi+j if −q ≤ i + j ≤ r and [gi, gj ] = 0 otherwise. Then g0
is a subalgebra of g, and each subspace gj is a g0-module under the adjoint
action. The spaces
g≤0 := g− ⊕ g0 and g≥0 := g0 ⊕ g+
are also subalgebras of g, where
g−i and g+ :=
are nilpotent ideals of g≤0 and g≥0 respectively. If g−q and gr are nonzero,
then q is said to be the depth and r the height of g. We assume that q, r ≥ 1
and q is finite, but in this section and the next allow the possibility that
the height r is infinite. The following conditions play a key role in this
(1.1) g−1 is an irreducible g0-module.
(1.2) If x ∈ g≥0 and [x, g−1] = 0, then x = 0.
Property (1.1) is termed irreducibility and (1.2) is called transitivity.
When we say an algebra is irreducible and transitive, we mean that both
(1.1) and (1.2) hold. On occasion we refer to algebras satisfying the following
constraint as being 1-transitive, or having 1-transitivity: