On the Derived Length of Lie Solvable Group Algebras

1.2 ASSOCIATED LIE ALGEBRA OF GROUP ALGEBRAS 5
Evidently, γ2(G) coincides with the commutator subgroup G ′ . For
H ⊆ G we will denote by CG(H) the centralizer of the subset H in G.
Furthermore, denote by Cn the cyclic group of order n and by Cg the
conjugacy class including the element g ∈ G.
The upper integral part of a real number r is denoted by ⌈r⌉.
1.2 Associated Lie algebra of group algebras
Let (L, +) be a vector space over the field F and assume that a
second binary operation [a, b] is defined in L and the identities
• α[a, b] = [αa, b] = [a, αb];
• [a + b, c] = [a, c] + [b, c] and [a, b + c] = [a, b] + [a, c];
• [a, a] = 0;
• [a, b], c + [b, c], a + [c, a], b = 0
hold for all a, b, c ∈ L and α ∈ F . Then we say that L is a Lie algebra
over the field F .
Let A be an associative algebra over the field F and x, y ∈ A. The
element [x, y] = xy − yx will be called the Lie commutator of x and y.
Let us introduce in A the new operation [x, y] = xy − yx. Then A is
a Lie algebra with respect to the operations + and [ , ], which is said
to be the associated Lie algebra of A.
For the sequence (xi) of elements of A we define the left n-normed
Lie commutator by induction as
[x1, x2, . . . , xn] =
[x1, x2, . . . , xn−1], xn .
We shall use freely the identities
[x, yz] = [x, y]z + y[x, z], [xy, z] = x[y, z] + [x, z]y,