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.