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