2 1. GRADED LIE ALGEBRAS
(1.3) If x g≤0 and [x, g1] = 0, then x = 0.
1.2. The Weisfeiler radical
Every graded Lie algebra g has a radical, which was first introduced by
Weisfeiler [W], and which is constructed as follows: Set
M0(g)
= 0, and
for i 0 define
Mi+1(g)
inductively by
(1.4)
Mi+1(g)
= {x g− | [x, g+]
Mi(g)}.
Then
(1.5)
0 =
M0(g)

M1(g)
···
Mi−1(g)

Mi(g)
. . . and
M(g) :=
i
Mi(g)
is called the Weisfeiler radical of g. By its definition, M(g) is a subspace
of g− invariant under bracketing by g+, and for j = 0, 1, . . . , q,
[[Mi(g),
g−j], g+]
[[Mi(g),
g+], g−j ] +
[Mi(g),
[g−j, g+]] (1.6)

[Mi−1(g),
g−j] +
[Mi(g),
≥1
g−j+ ].
Now when j = 0 and i = 1, the right side of (1.6) is zero, which implies that
[M1(g),
g0]
M1(g).
We may assume that
[Mi−1(g),
g0]
Mi−1(g).
Then
(1.6) shows that
[Mi(g),
g0]
Mi(g).
Suppose we know that
(1.7)
[Mi(g),
g−k]
Mi+k(g)
for all 0 k j. Then by (1.6) and induction on i we have,
[[Mi(g),
g−j ], g+]
[Mi−1(g),
g−j ] +
[Mi(g),
≥1
g−j+ ]
Mi+j−1(g).
Consequently,
[Mi(g),
g−j]
Mi+j(g)
for j = 0, 1, . . . . Thus M(g) is an ideal
of g, and it exhibits the following characteristics enjoyed by a “radical”:
Proposition 1.8.
(i) M(g) is a graded ideal of g contained in g−.
(ii) Suppose that g is irreducible (1.1) and transitive (1.2), and let J be
an ideal of g contained in g−. Then J M(g)

i≥2
g−i. Thus,
M(g) is the sum of all the ideals of g contained in g−. Moreover,
g/M(g) is irreducible and transitive, and M(g/M(g)) = 0.
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