On action of outer derivations on nilpotent ideals of Lie algebras
Action of outer derivations on nilpotent ideals of Lie algebras are considered. It is shown that for a nilpotent ideal I of a Lie algebra L over a field F the ideal I+D(I) is nilpotent, provided that charF=0 or I nilpotent of nilpotency class less than p−1, where p=charF. In particular, the sum N(...
Збережено в:
| Дата: | 2009 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2009
|
| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/153383 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On action of outer derivations on nilpotent ideals of Lie algebras / D.V. Maksimenko // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 74–82. — Бібліогр.: 6 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-153383 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1533832019-06-15T01:26:20Z On action of outer derivations on nilpotent ideals of Lie algebras Maksimenko, D.V. Action of outer derivations on nilpotent ideals of Lie algebras are considered. It is shown that for a nilpotent ideal I of a Lie algebra L over a field F the ideal I+D(I) is nilpotent, provided that charF=0 or I nilpotent of nilpotency class less than p−1, where p=charF. In particular, the sum N(L) of all nilpotent ideals of a Lie algebra L is a characteristic ideal, if charF=0 or N(L) is nilpotent of class less than p−1, where p=charF. 2009 Article On action of outer derivations on nilpotent ideals of Lie algebras / D.V. Maksimenko // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 74–82. — Бібліогр.: 6 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 17B40. http://dspace.nbuv.gov.ua/handle/123456789/153383 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Action of outer derivations on nilpotent ideals of Lie algebras are considered. It is shown that for a nilpotent ideal I of a Lie algebra L over a field F the ideal I+D(I) is nilpotent, provided that charF=0 or I nilpotent of nilpotency class less than p−1, where p=charF. In particular, the sum N(L) of all nilpotent ideals of a Lie algebra L is a characteristic ideal, if charF=0 or N(L) is nilpotent of class less than p−1, where p=charF. |
| format |
Article |
| author |
Maksimenko, D.V. |
| spellingShingle |
Maksimenko, D.V. On action of outer derivations on nilpotent ideals of Lie algebras Algebra and Discrete Mathematics |
| author_facet |
Maksimenko, D.V. |
| author_sort |
Maksimenko, D.V. |
| title |
On action of outer derivations on nilpotent ideals of Lie algebras |
| title_short |
On action of outer derivations on nilpotent ideals of Lie algebras |
| title_full |
On action of outer derivations on nilpotent ideals of Lie algebras |
| title_fullStr |
On action of outer derivations on nilpotent ideals of Lie algebras |
| title_full_unstemmed |
On action of outer derivations on nilpotent ideals of Lie algebras |
| title_sort |
on action of outer derivations on nilpotent ideals of lie algebras |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/153383 |
| citation_txt |
On action of outer derivations on nilpotent ideals of Lie algebras / D.V. Maksimenko // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 74–82. — Бібліогр.: 6 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT maksimenkodv onactionofouterderivationsonnilpotentidealsofliealgebras |
| first_indexed |
2025-07-14T04:36:28Z |
| last_indexed |
2025-07-14T04:36:28Z |
| _version_ |
1837595664078340096 |
| fulltext |
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2009). pp. 74 – 82
c© Journal “Algebra and Discrete Mathematics”
On action of outer derivations on nilpotent ideals
of Lie algebras
Dmitriy V. Maksimenko
Communicated by A. P. Petravchuk
Abstract. Action of outer derivations on nilpotent ideals of
Lie algebras are considered. It is shown that for a nilpotent ideal
I of a Lie algebra L over a field F the ideal I + D(I) is nilpotent,
provided that charF = 0 or I nilpotent of nilpotency class less
than p − 1, where p = charF . In particular, the sum N(L) of
all nilpotent ideals of a Lie algebra L is a characteristic ideal, if
charF = 0 or N(L) is nilpotent of class less than p − 1, where
p = charF .
It is known that the nilradical of a finite dimensional Lie algebra over
a field of characteristic 0 is characteristic, i.e. it is invariant under any
derivation of the algebra. It was shown in [3], that for an arbitrary Lie
algebra L (not necessarily finite dimensional) over a field of characteristic
0 the image D(I) of a nilpotent ideal I ⊆ L under derivation D ∈ Der(L)
lies in some nilpotent ideal of the algebra L. The restriction on charac-
teristic of the ground field is essential while proving this assertion.
We use methods which are analogous to ones in [6] during the investi-
gation of behavior of solvable ideals under outer derivations. It is shown
in Theorem 1 of the paper that the image of a nilpotent ideal of nilpotency
class n from a Lie algebra L over a field F under an outer derivation lies
in a nilpotent ideal provided that n < p−1, where p = charF. The meth-
ods of research here are completely different from ones in [3] because it is
impossible in general to construct automorphisms from nilpotent deriva-
tions of Lie algebras over fields of positive characteristic.
2000 Mathematics Subject Classification: 17B40.
Key words and phrases: Lie algebra, derivation, solvable radical, nilpotent
ideal.
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.D. Maksimenko 75
The notations in the paper are standard. If T is an F−subspace of a
Lie algebra L then we denote by T 1 = T, T 2 = [T, T ], . . . , Tn = [Tn−1, T ].
For elements x1, . . . , xn of a Lie algebra L we denote
[x1, x2, . . . , xn] = [[. . . [x1, x2], . . . xn−1], xn].
For a Lie algebra L we denote by Der(L) the Lie algebra of all deriva-
tions of L. If D ∈ Der(L) and T is an F -subspace of L we denote for
convenience D0(T ) = T, Dk(T ) = D(Dk−1(T )) for k > 1.
Further, for any elements x1, . . . , xm ∈ L, any derivation D ∈ Der(L)
and an arbitrary natural n > 1 it holds (Leibniz’s rule for differentiation
of several multipliers):
Dn([x1, . . . , xm]) =
∑
k1+···+km=n
n!
k1! . . . km!
[Dk1(x1), . . . , D
km(xm)] (1)
(the summation is extended over all nonnegative k1, . . . , km). The special
case of this formula is the usual Leibniz’s rule
Dn([x, y]) =
n∑
k=0
(
n
k
)
[Dk(x), Dn−k(y)]
for arbitrary elements x, y ∈ L и D ∈ Der(L).
Let L be a Lie algebra over an arbitrary field, let I be its ideal, D ∈
Der(L). As for any x ∈ I, y ∈ L it holds [x, D(y)] = D([x, y])− [D(x), y],
than I + D(I) is an ideal of the Lie algebra L. It is easy to see that the
sum
I + D(I) + D2(I) + . . . + Dk(I)
is also an ideal for any natural k > 1.
We need some lemmas for proving the main theorem.
Lemma 1. Let L be a Lie algebra over a field of characteristic p 6= 2, I
be an abelian ideal of L, D ∈ Der(L). Then [D(I), D(I)] ⊆ I.
Proof. Take arbitrary elements x, y ∈ I. As the ideal I is abelian, then
[x, y] = 0 and, therefore, D2([x, y]) = 0. From the other hand, by Leib-
niz’s rule we obtain the following:
0 = D2([x, y]) = [D2(x), y] + 2[D(x), D(y)] + [x, D2(y)].
Since I is an ideal of L, it follows from the previous relation that
[D(x), D(y)] ∈ I. Since x, y are arbitrary elements from I then
[D(I), D(I)] ⊆ I.
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.76 On action of outer derivations on nilpotent ideals
Lemma 2. Let L be a Lie algebra over an arbitrary field, I be an ideal
of L, D ∈ Der(L). Then for any x1, . . . , xs ∈ I and any nonnegative
number m < s it holds:
Dm([x1, . . . , xs]) ∈ Is−m.
Proof. Denote by l = s − m > 0. Using the relation (1), we obtain:
Dm([x1, . . . , xs]) =
∑
k1+···+ks=m
m!
k1! . . . ks!
[Dk1(x1), . . . , D
ks(xs)] (2)
Since all k1, . . . , ks are nonnegative and k1 + · · · + ks = m < s, then
at least l of the numbers k1, . . . , ks are equal to zero. As, by definition,
D0(x) = x for all x ∈ I, then, as one can easily make sure, every sum-
mand [Dk1(x1), D
k2(x2), . . . , D
ks(xs)] of this sum belongs to I l. Hence,
Dm([x1, . . . , xs]) ∈ I l = Is−m.
Lemma 3. Let I be a nilpotent ideal of nilpotency class n from a Lie
algebra L over a field K of characteristic 0 or characteristic p > n + 1,
D ∈ Der(L). Then (I + D(I))n+1 ⊆ I.
Proof. To prove the statement of Lemma it is sufficient to show that
[D(I), . . . , D(I)
︸ ︷︷ ︸
n+1
] ⊆ I (3)
Consider the equality (2) for m = n + 1, s = n + 1 and take into
account that [x1, . . . , xn, xn+1] = 0 for all elements x1, . . . , xn, xn+1 ∈ I:
Dn+1([x1, . . . , xn+1]) =
=
∑
k1+···+kn+1=n+1
(n + 1)!
k1! . . . k(n+1)!
[Dk1(x1), . . . , D
kn+1(xn+1)] = 0.
Since all k1, . . . , kn+1 are nonnegative, then the last relation can be writ-
ten down in the form
(n + 1)!
1! . . . 1!
[D(x1), . . . , D(xn+1)]+
+
∑
k1+···+kn+1=n+1
(n + 1)!
k1! . . . kn+1!
[Dk1(x1), . . . , D
kn+1(xn+1)] = 0,
where the summation is extended over all nonnegative k1, . . . , kn+1, at
least one of which is more then 1. Since all numbers k1, . . . , kn+1 in the
last sum are nonnegative, then at least, one of them is zero. Therefore all
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.D. Maksimenko 77
summands under the sign of sum in the last relation belong to the ideal
I. But then, obviously, (n + 1)![D(x1), . . . , D(xn+1)] ∈ I. As n + 1 < p
or charF = 0, then it follows that [D(x1), . . . , D(xn+1)] ∈ I. Since
the elements x1, . . . , xn, xn+1 ∈ I were chosen arbitrarily we obtain the
relation (3).
Lemma 4. Let I be a nilpotent ideal of nilpotency class n from a Lie
algebra L over a field of characteristic 0 or characteristic p > n + 1,
D ∈ Der(L). Then [I, D(I), . . . , D(I)
︸ ︷︷ ︸
n+1
] ⊆ I2.
Proof. Take arbitrary elements x1, . . . , xn+2 ∈ I. Denote for convenience:
t1 = [x1, D(x2), D(x3), . . . , D(xn+2)];
t2 = [D(x1), x2, D(x3), . . . , D(xn+2)];
. . .
tn+1 = [D(x1), D(x2), . . . , xn+1, D(xn+2)];
tn+2 = [D(x1), D(x2), . . . , D(xn+1), xn+2].
Since In+1 = 0, we can write down the following equalities:
us = [x1, x2, . . . , D(xs), . . . , xn+2] = 0
for s = 1, . . . , n + 2. Applying the Leibniz’s rule (1) for computation
0 = Dn(us) = Dn([x1, x2, . . . , D(xs), . . . , xn+2]) we obtain
Dn(us) =
∑
k1+···+kn+2=n
n!
k1! . . . kn+2!
[Dk1(x1), . . .
. . . , Dks+1(xs), . . . , D
kn+2(xn+2)].
Since all kj are nonnegative and k1 + · · · + kn+2 = n, then at least two
numbers among k1, . . . , kn+2 are equal to 0. If at least three numbers
among k1, . . . , kn+2 are equal to 0 then the summand of this sum of the
form
n!
k1! . . . kn+2!
[Dk1(x1), . . . D
ks+1(xs), . . . , D
kn+2(xn+2)]
lies obviously in I2. Let now exactly two numbers ki, kj are equal to 0 in
this summand. If i 6= s и j 6= s, then, as above, one can show that the
summand
n!
k1! . . . kn+2!
[Dk1(x1), . . . D
ks+1(xs), . . . , D
kn+2(xn+2)]
lies in the ideal I2. So, we have to consider only the case when one of
the indices i, j, for instance, i coincides with s. Then ks = 0, kj = 0, j 6=
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.78 On action of outer derivations on nilpotent ideals
s. Since all other numbers km are equal to 1, then we obtain that the
summand
n!
k1! . . . kn+2!
[Dk1(x1), . . . D
ks+1(xs), . . . , D
kn+2(xn+2)]
is equal to
n!
1! . . . 1!
[D(x1), . . . D(xj−1), D
0(xj), D(xj+1) . . . , Dkn+2(xn+2)] = n!tj .
Therefore, having fixed i = s and arbitrarily chosen j, not equal to s, we
obtain that
Dn(us) = n!(t1 + . . . ts−1 + ts+1 + · · · + tn+2) + zs (4)
for some zs ∈ I2. Denote by vs = Dn(us)/n! for s = 1, . . . , n + 2. Then
taking into account the relation charK = p > n + 1 we see that
vs = t1 + . . . ts−1 + ts+1 + · · · + tn+2 ∈ I2
for arbitrary s = 1, . . . , n + 2. Consider the sum v =
∑n+2
s=1 vs. It is easy
to see that v = (n + 1)
∑n+2
k=1 tk, v ∈ I2. Because of the restriction on
characteristic of the ground field it holds the relation t = t1 + t2 + · · · +
tn+2 ∈ I2. But then the element t1 = t − v1 belongs to the ideal I2. As
elements x1, . . . , xn+2 were chosen arbitrarily and
t1 = [x1, D(x2), D(x3), . . . , D(xn+2)]
we have that [I, D(I), . . . , D(I)
︸ ︷︷ ︸
n+1
] ⊆ I2.
Lemma 5. Let I be a nilpotent ideal of nilpotency class n from a Lie
algebra L over a field of characteristic 0 or characteristic p > n + 1,
D ∈ Der(L). Then there exits a function fn(m) of a natural argument
m such that fn(m) = fn(m − 1) + n − m + 1, fn(1) = n + 1 and
[Im, D(I), . . . , D(I)
︸ ︷︷ ︸
fn(m)
] ⊆ Im+1 (5)
for m = 1, . . . , n.
Proof. Let n be a fixed natural number. Then for m = 1 we have by
Lemma 4 the relation [I, D(I), . . . , D(I)
︸ ︷︷ ︸
n+1
] ⊆ I2 and therefore one can
take fn(1) = n + 1.
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.D. Maksimenko 79
Assume that it is already proved that the function fn(t) satisfies the
condition
[Im−1, D(I), . . . , D(I)
︸ ︷︷ ︸
fn(m−1)
] ⊆ Im.
Let us show that the following inclusion holds:
[Im, D(I), . . . , D(I)
︸ ︷︷ ︸
fn(m−1)+n−m+1
] ⊆ Im+1.
We denote for convenience N = fn(m−1)+n−m+2 and take arbitrary
elements x1 ∈ Im, x2, . . . , xN ∈ I. Denote by s = fn(m − 1) + 1, t =
n − m + 1. Then N = t + s.
It is easy to see that the following equality holds:
[x1, D(x2), . . . , D(xs), xs+1, . . . , xN ] = 0 (6)
Really, [x1, D(x2), . . . , D(xs)] ∈ Im and, as xs+1, . . . , xN ∈ I, then
[x1, D(x2), . . . , D(xs), xs+1, . . . , xN
︸ ︷︷ ︸
n−m+1
] ∈ Im+(n−m+1) = In+1 = 0.
Apply now the derivation D to the equality (6) n−m+1 times. Using
Leibniz’s rule (1), we obtain:
∑ t!
k1! . . . kN !
[Dk1(x1), D
k2+1(x2), . . .
. . . , Dks+1(xs), D
ks+1(xs+1), . . . , D
kN (xN )] = 0 (7)
where the summation is extended over all nonnegative k1, . . . , kN such
that k1 + · · · + kN = t = n − m + 1.
Since the sum of all numbers k1, . . . , kN is t, and their quantity is
N = s + t, then obviously there are at least s numbers from the set
{k1, . . . , kN} which are equal to 0. Let’s prove that in the sum (7) all
summands except maybe the summand
t![D0(x1), D(x2), . . . , D(xs), D(xs+1), . . . , D(xN )], (8)
that corresponds to k1 = 0, k2 = 0, . . . , ks = 0, ks+1 = 1, . . . , kN = 1, lie
in Im+1.
Consider the possible cases:
a) There are exactly s numbers among k1, . . . , kN which are equal
to 0. If these numbers are k1, . . . , ks, then ks+1 = · · · = kN = 1 and
we obtain the exceptional element (8). So we assume that at least one
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.80 On action of outer derivations on nilpotent ideals
of the numbers k1, . . . , ks is nonzero. Then at least one of the numbers
ks+1, . . . , kN is 0.
At first assume that k1 = 0. Then Dk1(x1) = x1 ∈ Im and if at least
one of numbers ks+1, . . . , kN is 0, then the summand
t! · [Dk1(x1), D
k2+1(x2), . . . , D
ks+1(xs), D
ks+1(xs+1), . . . , D
kN (xN )] (9)
belongs to the ideal Im+1. Let now all the numbers ks+1, . . . , kN be
nonzero. Then k2 = · · · = ks = 0 and we obtain the exceptional ele-
ment (8).
Consider now the case k1 = 1. If k2 = · · · = ks = 0, then D(x1) ∈
Im−1 by Lemma 2 and therefore [D(x1), D(x2), . . . , D(xs)
︸ ︷︷ ︸
fn(m−1)
] ∈ Im. Since
at least one of the numbers ks+1, . . . , kN is 0, then the element of the
form (9) lies in Im+1. Suppose now that at least one of the numbers
k2, . . . , ks is equal to 1. Then at least two of the numbers ks+1, . . . , kN
are 0 and therefore again the element of the form (9) lies in Im+1.
So, in case a) either the element (9) is of the exceptional form (8) or
it lies in Im+1.
b) There are exactly s+ i numbers among k1, . . . , kN which are equal
to 0, where i > 1. Show that we can suppose in this case that at least
i + 1 of the numbers ks+1, . . . , kN are equal to 0. Really, since N = s + t
then we have that at least i of numbers ks+1, . . . , kN are equal to 0.
Assume that there are exactly i such numbers. Then all the numbers
k1, . . . , ks are equal to 0 and therefore t! · [x1, D(x2), . . . , D(xs)] ∈ Im.
Since i > 1, then at least one of the numbers ks+1, . . . , kN is equal to 0
and t! · [x1, D(x2), . . . , D(xs), D
ks+1(xs+1), . . . , D
kN (xN )] ∈ Im+1.
So, we will suppose further that there are at least i+1 of the numbers
ks+1, . . . , kN which are equal to 0. Denote the quantity of such numbers
by r. Then according to our assumption r > i + 1. Hence, the quantity
of non-zero numbers among ks+1, . . . , kN is equal to t − r and for their
sum it holds > t − r. But then the sum of all non-zero numbers among
k1, . . . , ks is less or equal t − (t − r) = r and, therefore k1 6 r.
At first let the sum of all nonzero numbers among k1, . . . , ks be less
than r. Then k1 6 r−1 and therefore Dk1(x1) ∈ Im−r+1. It follows from
here that
[Dk1(x1), D
k2+1(x2), . . .
. . . , Dks+1(xs), D
ks+1(xs+1), . . . , D
kN (xN )] ∈ Im−r+1+r = Im+1
since there are at least r elements among Dks+1(xs+1), . . . , D
kN (xN )
which lie in I.
Let now the sum of all nonzero numbers among k1, . . . , ks be equal
to r. If k1 6 r − 1, then, as above, one can show that the element of
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.D. Maksimenko 81
the form (8) lies in Im+1. Let now k1 = r. Then k2 = · · · = ks = 0 and
by the inductive assumption (since by Lemma 2 it holds Dr(x1) ∈ Im−r)
we have the inclusion [Dr(x1), D(x2), . . . , D(xs)
︸ ︷︷ ︸
fn(m−1)
] ∈ Im−r+1. But then
[Dr(x1), D(x2), . . . , D(xs), D
ks+1(xs+1), . . . , D
kN (xN )] ∈ Im+1, since at
least r elements among Dks+1(xs+1), . . . , D
kN (xN ) belong to I.
So, all summands in the relation (7), except maybe of the form (8)
lie in Im+1. But then in view of equality (7) the exceptive summand (8)
lies in Im+1. As the characteristic of the ground field does not divide
t = n − m + 1, we obtain [x1, D(x2), . . . , D(xN )] ∈ Im+1. Since the
elements x1 ∈ Im, x2, . . . , xN ∈ I can be chosen arbitrarily we obtain
[Im, D(I), . . . , D(I)
︸ ︷︷ ︸
fn(m−1)+n−m+1
] ⊆ Im+1
It means that one can put fn(m) = fn(m − 1) + n − m + 1. Lemma is
proved.
Remark 1. The relation for the function fn(m) obtained while proving
the previous lemma is an inhomogeneous recurrence relation of the 1-st
order. Its solution (see, for example [2], §3.3.3) can be written down as
a sum fn = fh
n +fp
n, where fh
n is the general solution of the homogeneous
recurrence relation fn(m)−fn(m−1) = 0, and fp
n is a particular solution
for the inhomogeneous relation
fn(m) = fn(m − 1) + n − m + 1 (10)
The single characteristic root of the corresponding homogeneous relation
is 1. So, its general solution is fh
n = C, where C is an arbitrary constant.
We find a particular solution in the form fp
n = m(A1m+A0), where A0, A1
are indeterminate coefficients. Substituting fp
n into the relation (10), we
get A1 = −1
2 , A0 = n + 1
2 . So, the general solution of the inhomogeneous
relation (10) can be presented as fn(m) = C− 1
2m2 +(n+ 1
2)m. One finds
the coefficient C = 1 from the initial condition fn(1) = n + 1. Finally,
we have fn(m) = m(n + 1) − (m − 1)(m + 2)/2.
Theorem 1. Let I be a nilpotent ideal of nilpotency class n of a Lie
algebra L over a field of characteristic 0 or characteristic p > n + 1,
D ∈ Der(L). Then I + D(I) is a nilpotent ideal of the Lie algebra L of
nilpotency class at most n(n + 1)(2n + 1)/6 + 2n.
Proof. Denote by k =
∑n
m=1 fn(m). Using Lemma 5 one can easily show
that [I, D(I), . . . , D(I)
︸ ︷︷ ︸
k
] ⊆ In+1 = 0. Further, by Lemma 3 we have
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.82 On action of outer derivations on nilpotent ideals
(I + D(I))k+n+1 = 0. So, the ideal I + D(I) is nilpotent of nilpotency
class at most k + n. Direct calculation yields k + n = n +
∑n
m=1 m(n +
1) −
∑n
m=1(m − 1)(m + 2)/2 = n(n + 1)(2n + 1)/6 + 2n.
Corollary 1. Let L be a Lie algebra (not necessarily finite dimensional)
over a field F, let N(L) be the sum of all nilpotent ideals of L. If the ideal
N(L) is nilpotent, then it is a characteristic in the following cases: a)
charF = 0; b) charF = p > 0 and nilpotency class of N(L) is less than
p − 1.
Remark 2. We should note that the estimation of nilpotency class of
the ideal I + D(I) from Theorem 1 is rather rough. For example, for an
ideal I of nilpotency class 2 of a Lie algebra over a field of characteristic
p > 3 Theorem 1 gives the estimation 9, but direct calculation shows that
nilpotency class of I + D(I) does not exceed 8.
References
[1] N.A. Jacobson, Lie algebras . Interscience tracts, no. 10 (New York, 1962).
[2] Kenneth H. Rosen (Editor-in-Chief), Handbook of Discrete and Combinatorial
Mathematics, CRC Press, (2000).
[3] B. Hartley, Locally nilpotent ideals of a Lie algebra, Proc. Cambridge Phil. Soc.,
63 (1967), 257–272.
[4] G. Letzter, Derivations and nil ideals , Rendicotti del Circolo Matematico di
Palermo, 37, no.2 (1988), 174-176.
[5] V.S.Luchko, A.P.Petravchuk, On one-sided ideals of associative rings, Algebra
and Discrete Mathematics, no.3 (2007), P.1-6.
[6] A.P.Petravchuk, On behavior of solvable ideals of Lie algebras under outer deriva-
tions, Communications in Algebra, to appear.
Contact information
Dmitriy V.
Maksimenko
Kiev Taras Shevchenko University, Fac-
ulty of Mechanics and Mathematics, 64,
Volodymyrska street, 01033 Kyiv, Ukraine
E-Mail: dm.mksmk@gmail.com
Received by the editors: 24.09.2007
and in final form 14.04.2009.
|