Introduction

The focus of this work is the following Main Theorem, often referred to

as the “Recognition Theorem,” because of its extensive use in recognizing

certain graded Lie algebras from their null components.

Theorem 0.1. Let g =

r

j=−q

gj be a finite-dimensional graded Lie

algebra over an algebraically closed field F of characteristic p 3. Assume

that:

(a) g0 is a direct sum of ideals, each of which is abelian, a classical

simple Lie algebra, or one of the Lie algebras gln, sln, or pgln with

p | n;

(b) g−1 is an irreducible g0-module;

(c) If x ∈

j≥0

gj and [x, g−1] = 0, then x = 0;

(d) If x ∈

j≥0

g−j and [x, g1] = 0, then x = 0.

Then g is isomorphic as a graded Lie algebra to one of the following:

(1) a classical simple Lie algebra with a standard grading;

(2) pglm for some m such that p | m with a standard grading;

(3) a Cartan type Lie algebra with the natural grading or its reverse;

(4) a

Melikyan1

algebra (in characteristic 5) with either the natural

grading or its reverse.

The classical simple Lie algebras in this theorem are the algebras ob-

tained by reduction modulo p of the finite-dimensional complex simple Lie

algebras, as described in [S, Sec. 10] (see also Section 2.2). Thus, they are

of type

An−1, p | n, Bn, Cn, Dn, E6, E7, E8, F4, G2, or they are isomorphic

to psln where p | n.

The Recognition Theorem is an essential ingredient in the classification

of the finite-dimensional simple Lie algebras over algebraically closed fields

of characteristic p 3. In a sense, the whole classification theory is built

around this theorem, as the theory aims to show that any finite-dimensional

simple Lie algebra L admits a filtration L = L−q ⊃ . . . ⊃ L0 ⊃ ··· ⊃ Lr ⊃

Lr+1 = 0 such that the corresponding graded Lie algebra g =

r

i=−q

gi,

where gi = Li/Li+1, satisfies conditions (a)-(d) above. The Recognition

Theorem is used several times throughout the classification; its first appli-

cation results in a complete list of the simple Lie algebras of absolute toral

1transliterated

as Melikian in many references such as [St], for example

ix