On the Derived Length of Lie Solvable Group Algebras

1.2 ASSOCIATED LIE ALGEBRA OF GROUP ALGEBRAS 7
called the Lie derived length of A. Similarly, the algebra A is said to
be strongly Lie solvable of derived length dl L (A) = m if δ (m) (A) = 0
and δ (m−1) (A) = 0.
Let a and b be elements of the unit group U(A) of A. Then by the
equality (a, b) = 1 + a −1 b −1 [a, b] the m-th term of the derived series
U(A) is contained in 1 + δ (m) (A). Thus to the investigation of the
derived length of U(A) the strong Lie derived series is a useful tool.
Now we establish further two series, which can be applied similarly for
the study of the nilpotency class of U(A).
Let A [1] = A and for n > 1 let A [n] be the ideal of A generated by
all the Lie commutators [x1, . . . , xn] with x1, . . . , xn ∈ A. Then the
ideal A [n] is the n-th lower Lie power and the series
A = A [1] =⊇ A [2] ⊇ · · · ⊇ A [n] ⊇ · · ·
is called the lower Lie power series of the associative algebra A.
By induction, we define the n-th upper Lie power A (n) of A as the
ideal generated by all the Lie commutators [x, y], where x ∈ A (n−1) ,
y ∈ A and A (1) = A. The series
A = A (1) ⊇ A (2) ⊇ · · · ⊇ A (n) ⊇ · · ·
is the upper Lie power series of the associative algebra A. Clearly,
A [n] ⊆ A (n) for all n.
The algebra A is called Lie nilpotent if there exists n such that
A [n] = 0 and the least integer of this kind is called the Lie nilpotency
index of A and it is denoted by tL(A). Similarly, A is said to be
upper Lie nilpotent and its upper Lie nilpotency index is t L (A) = m if
A (m) = 0 but A (m−1) = 0.
Evidently, δ (0) (A) = A (1) and assume that δ (n) (A) ⊆ A (2n ) for some
n ≥ 0. By the well-known inclusion [A (i) , A (j) ] ⊆ A (i+j) we obtain
δ (n+1) (A) = δ (n) (A), δ (n) (A) A ⊆ A (2n ) , A (2 n ) A
We have just shown that
⊆ A (2n+1 ) A = A (2 n+1 ) .
δ (n) (A) ⊆ A (2n )
for all n ≥ 0,
which is an analog of a standard fact from group theory.