CHAPTER 1

Graded Lie Algebras

1.1. Introduction

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

Theorem.

To begin, our focus is on Lie algebras over an arbitrary field F having

an integer grading,

g =

r

i=−q

gi,

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− :=

q

i=1

g−i and g+ :=

r

j=1

gj

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

investigation:

(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:

1