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
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