On nilpotent Lie algebras of derivations with large center
Let K be a field of characteristic zero and A an associative commutative K-algebra that is an integral domain. Denote by R the quotient field of A and by W(A)=RDerA the Lie algebra of derivations on R that are products of elements of R and derivations on A. Nilpotent Lie subalgebras of the Lie algeb...
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| Cite this: | On nilpotent Lie algebras of derivations with large center / K. Sysak // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 153-162. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1552122019-06-17T01:26:21Z On nilpotent Lie algebras of derivations with large center Sysak, K. Let K be a field of characteristic zero and A an associative commutative K-algebra that is an integral domain. Denote by R the quotient field of A and by W(A)=RDerA the Lie algebra of derivations on R that are products of elements of R and derivations on A. Nilpotent Lie subalgebras of the Lie algebra W(A) of rank n over R with the center of rank n−1 are studied. It is proved that such a Lie algebra L is isomorphic to a subalgebra of the Lie algebra un(F) of triangular polynomial derivations where F is the field of constants for L. 2016 Article On nilpotent Lie algebras of derivations with large center / K. Sysak // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 153-162. — Бібліогр.: 8 назв. — англ. 1726-3255 2010 MSC:Primary 17B66; Secondary 17B30, 13N15. http://dspace.nbuv.gov.ua/handle/123456789/155212 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let K be a field of characteristic zero and A an associative commutative K-algebra that is an integral domain. Denote by R the quotient field of A and by W(A)=RDerA the Lie algebra of derivations on R that are products of elements of R and derivations on A. Nilpotent Lie subalgebras of the Lie algebra W(A) of rank n over R with the center of rank n−1 are studied. It is proved that such a Lie algebra L is isomorphic to a subalgebra of the Lie algebra un(F) of triangular polynomial derivations where F is the field of constants for L. |
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On nilpotent Lie algebras of derivations with large center |
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on nilpotent lie algebras of derivations with large center |
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On nilpotent Lie algebras of derivations with large center / K. Sysak // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 153-162. — Бібліогр.: 8 назв. — англ. |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 1, pp. 153–162
© Journal “Algebra and Discrete Mathematics”
On nilpotent Lie algebras of derivations
with large center
Kateryna Sysak
Communicated by A. P. Petravchuk
Abstract. Let K be a field of characteristic zero and A
an associative commutative K-algebra that is an integral domain.
Denote by R the quotient field of A and by W (A) = RDerA the
Lie algebra of derivations on R that are products of elements of R
and derivations on A. Nilpotent Lie subalgebras of the Lie algebra
W (A) of rank n over R with the center of rank n− 1 are studied. It
is proved that such a Lie algebra L is isomorphic to a subalgebra of
the Lie algebra un(F ) of triangular polynomial derivations where
F is the field of constants for L.
Introduction
Let K be an algebraically closed field of characteristic zero, and A be an
associative commutative algebra over K with identity, without zero divisors.
A K-linear mapping D : A −→ A is called K-derivation of A if D satisfies
the Leibniz’s rule: D(ab) = D(a)b+ aD(b) for all a, b ∈ A. The set DerA
of all K-derivations on A forms a Lie algebra over K with respect to the
operation [D1, D2] = D1D2 −D2D1, D1, D2 ∈ DerA. Denote by R the
quotient field of the integral domain A. Each derivation D of A is uniquely
extended to a derivation of R by the rule: D(a/b) = (D(a)b− aD(b))/b2.
Denote by DerR the Lie algebra (over K) of all K-derivations on R.
2010 MSC: Primary 17B66; Secondary 17B30, 13N15.
Key words and phrases: derivation, Lie algebra, nilpotent Lie subalgebra, tri-
angular derivation, polynomial algebra.
154 On nilpotent Lie algebras of derivations
Define the mapping rD : R −→ R by (rD)(s) = r ·D(s) for all r, s ∈ R.
It is easy to see that rD is a derivation of R. The R-linear hull of the
set {rD|r ∈ R,D ∈ DerA} forms the vector space RDerA over R, which
is a Lie subalgebra of DerR. Observe that RDerA is a Lie algebra over
K but not always over R, and DerA is embedded in a natural way into
RDerA. Many authors study the Lie algebra of derivations DerA and
its subalgebras, see [2–7].
This paper deals with nilpotent Lie subalgebras of the Lie algebra
RDerA. Let L be a Lie subalgebra of RDerA. The subfield F = F (L)
of R consisting of all a ∈ R such that D(a) = 0 for all D ∈ L is called
the field of constants for L. Let us denote by RL the R-linear hull of L
and, analogously, by FL the linear hull of L over its field of constants
F = F (L). The rank of L over R is defined as the dimension of the vector
space RL over R, i.e. rankR L = dimR RL.
The main results of the paper:
• (Theorem 1) If L is a nilpotent Lie subalgebra of the Lie algebra
RDerA of rank n over R such that its center Z(L) is of rank n− 1
over R and F is the field of constants for L, then the Lie algebra
FL is contained in the Lie subalgebra of RDerA of the form
F
〈
D1, aD1,
a2
2!
D1, . . . ,
as
s!
D1, D2, aD2, . . . ,
as
s!
D2, . . . ,
Dn−1, . . . ,
as
s!
Dn−1, Dn
〉
,
where D1, D2, . . . , Dn ∈ FL are such that [Di, Dj ] = 0, i, j =
1, . . . , n, and a ∈ R is such that D1(a) = D2(a) = · · · = Dn−1(a) =
0 and Dn(a) = 1.
• (Theorem 2) The Lie algebra FL is isomorphic to some subalgebra
of the Lie algebra un(F ) of triangular polynomial derivations.
Recall that the Lie algebra un(K) of triangular polynomial derivations
consists of all derivations of the form
D = f1(x2, . . . , xn)
∂
∂x1
+ f2(x3, . . . , xn)
∂
∂x2
+ · · ·
+ fn−1(xn)
∂
∂xn−1
+ fn
∂
∂xn
,
where fi ∈ K[xi+1, . . . , xn], i = 1, . . . , n − 1, and fn ∈ K. It is a Lie
subalgebra of the Lie algebra Wn(K) of all K-derivations on the polyno-
mial algebra K[x1, . . . , xn]. Such subalgebras are studied in [2, 3]. As Lie
algebras, they are locally nilpotent but not nilpotent.
K. Sysak 155
We use the standard notations. The Lie algebra RDerA is denoted by
W (A), as in [7]. The linear hull of elements D1, D2, . . . , Dn over the field K
is denoted by K〈D1, D2, . . . , Dn〉. If L is a Lie subalgebra of a Lie algebra
M , then the normalizer ofL inM is the setNM (L) = {x ∈ M | [x, L] ⊆ L}.
Obviously, NM (L) is the largest subalgebra of M in which L is an ideal.
1. Nilpotent Lie subalgebras of R Der A with the center
of large rank
We use Lemmas 1-5 proved in [7].
Lemma 1 ([7, Lemma 1]). Let D1, D2 ∈ W (A) and a, b ∈ R. Then
(a) [aD1, bD2] = ab[D1, D2] + aD1(b)D2 − bD2(a)D1.
(b) If a, b ∈ KerD1 ∩ KerD2, then [aD1, bD2] = ab[D1, D2].
Lemma 2 ([7, Lemma 2]). Let L be a nonzero Lie subalgebra of the Lie
algebra W (A), and F be the field of constants for L. Then FL is a Lie
algebra over F , and if L is abelian, nilpotent or solvable, then the Lie
algebra FL has the same property.
Lemma 3 ([7, Theorem 1]). Let L be a nilpotent Lie subalgebra of finite
rank over R of the Lie algebra W (A), and F be the field of constants for
L. Then FL is finite dimensional over F .
Lemma 4 ([7, Lemma 4]). Let L be a Lie subalgebra of the Lie algebra
W (A), and I be an ideal of L. Then the vector space RI ∩ L over K is
an ideal of L.
Lemma 5 ([7, Lemma 5]). Let L be a nilpotent Lie subalgebra of rank
n > 0 over R of the Lie algebra W (A). Then:
(a) L contains a series of ideals
0 = I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ In = L
such that rankR Ik = k for each k = 1, . . . , n.
(b) L possesses a basis D1, . . . , Dn over R such that Ik = (RD1 + · · · +
RDk) ∩ L, k = 1, . . . , n, and [L,Dk] ⊂ Ik−1.
(c) dimF FL/FIn−1 = 1.
Lemma 6. Let L be a nilpotent Lie subalgebra of the Lie algebra W (A),
and F be the field of constants for L. If L is of rank n > 0 over R and
its center Z(L) is of rank n− 1 over R, then L contains an abelian ideal
I such that dimF FL/FI = 1.
156 On nilpotent Lie algebras of derivations
Proof. Since the center Z(L) is of rank n− 1 over R, we can take linearly
independent over R elements D1, D2, . . . , Dn−1 ∈ Z(L). Let us consider
I = RZ(L) ∩ L = (RD1 + · · · +RDn−1) ∩ L.
In view of Lemma 4, I is an ideal of the Lie algebra L. Let us show that
I is an abelian ideal.
We first show that for an arbitrary element D = r1D1 + r2D2 +
· · · + rn−1Dn−1 ∈ I, its coefficients r1, r2, . . . , rn−1 ∈
n−1⋂
i=1
KerDi. For each
Di ∈ Z(L), i = 1, . . . , n− 1, let us consider
[Di, D] = [Di, r1D1 + r2D2 + · · · + rn−1Dn−1] =
n−1∑
j=1
[Di, rjDj ].
By Lemma 1, [Di, rjDj ] = rj [Di, Dj ] + Di(rj)Dj = Di(rj)Dj . Since
Di ∈ Z(L), we get
[Di, D] =
n−1∑
j=1
Di(rj)Dj = 0.
This implies that Di(r1) = Di(r2) = · · · = Di(rn−1) = 0 because
D1, D2, . . . , Dn−1 ∈ L are linearly independent over R. Therefore, rj ∈
n−1⋂
i=1
KerDi for j = 1, . . . , n− 1.
Now we take arbitrary D,D′ ∈ I and show that [D,D′] = 0. Let
D = a1D1+a2D2+· · ·+an−1Dn−1 andD′ = b1D1+b2D2+· · ·+bn−1Dn−1.
Then
[D,D′] =
n−1∑
i,j=1
(aibj [Di, Dj ] + aiDi(bj)Dj − bjDj(ai)Di) = 0
since ai, bj ∈
n−1⋂
i=1
KerDi for all i, j = 1, . . . , n − 1, and I is an abelian
ideal.
It is easy to see that FI is an abelian ideal of the Lie algebra FL over
F and dimF FL/FI = 1 in view of Lemma 5(c).
Remark 1. It follows from the proof of Lemma 6 that for an arbitrary
D = a1D1 + a2D2 + · · · + an−1Dn−1 ∈ FI, the inclusions ai ∈
n−1⋂
k=1
KerDk
hold for i = 1, . . . , n− 1.
K. Sysak 157
Lemma 7. Let L be a Lie subalgebra of rank n over R of the Lie algebra
W (A), {D1, D2, . . . , Dn} be a basis of L over R, and F be the field of
constants for L. Let there exists a ∈ R such that D1(a) = D2(a) = · · · =
Dn−1(a) = 0 and Dn(a) = 1. Then if b ∈ R satisfies the conditions
D1(b) = D2(b) = · · · = Dn−1(b) = 0 and Dn(b) ∈ F 〈1, a, . . . , as
s! 〉 for
some integer s > 0, then b ∈ F 〈1, a, . . . , as
s! ,
as+1
(s+1)!〉.
Proof. Since Dn(b) ∈ F 〈1, a, . . . , as
s! 〉, the equality Dn(b) =
s∑
i=0
βi
ai
i! holds
for some βi ∈ F , i = 0, . . . , s. Take an element c =
s∑
i=0
βi
ai+1
(i+1)! from R.
It is easy to check that D1(c) = D2(c) = · · · = Dn−1(c) = 0, because
D1(a) = D2(a) = · · · = Dn−1(a) = 0 by the conditions of the lemma.
Since Dn(a) = 1, the equality Dn(c) =
s∑
i=0
βi
ai
i! = Dn(b) holds, and so
Dk(b− c) = 0 for all k = 1, . . . , n. This implies that b− c ∈ F , hence for
some γ ∈ F , we obtain
b = γ + c = γ +
s∑
i=0
βi
ai+1
(i+ 1)!
.
Thus,
b ∈ F
〈
1, a, . . . ,
as
s!
,
as+1
(s+ 1)!
〉
,
and the proof is complete.
Theorem 1. Let L be a nilpotent Lie subalgebra of the Lie algebra W (A),
and let F = F (L) be the field of constants for L. If L is of rank n and its
center Z(L) is of rank n−1 over R, then there exist D1, D2, . . . , Dn ∈ FL,
a ∈ R, and an integer s > 0 such that FL is contained in the Lie subalgebra
of W (A) of the form
F
〈
D1, aD1,
a2
2!
D1, . . . ,
as
s!
D1, D2, aD2, . . . ,
as
s!
D2, . . . ,
Dn−1, . . . ,
as
s!
Dn−1, Dn
〉
,
where [Di, Dj ] = 0 for i, j = 1, . . . , n, Dn(a) = 1, and D1(a) = D2(a) =
· · · = Dn−1(a) = 0.
158 On nilpotent Lie algebras of derivations
Proof. It is easy to see that the vector space over F of the form
F
〈
D1, aD1,
a2
2!
D1, . . . ,
as
s!
D1, D2, aD2, . . . ,
as
s!
D2, . . . ,
Dn−1, . . . ,
as
s!
Dn−1, Dn
〉
is a Lie algebra over F . We denote it by L̃.
By Lemma 6, the Lie algebra L contains an abelian ideal I such that
FI is of codimension 1 in FL over F . The ideal I contains an R-basis
{D1, D2, . . . , Dn−1} of the center Z(L). Let us take an arbitrary element
Dn ∈ L that is not in Z(L). Then {D1, D2, . . . , Dn−1, Dn} is an R-basis
of L and FL = FI + FDn, where FI is an abelian ideal of FL.
Let us consider the action of the inner derivation adDn on the vector
space FI. It is easy to see that dimF Ker(adDn) = n− 1. Indeed, let
D = r1D1 + r2D2 + · · · + rn−1Dn−1 ∈ Ker(adDn).
Then
[Dn, D] =
n−1∑
i=1
[Dn, riDi] =
n−1∑
i=1
Dn(ri)Di = 0
whence Dn(ri) = 0 for all i = 1, . . . , n− 1.
By Remark 1, r1, r2, . . . , rn−1 ∈ F . Thus, Ker(adDn) =
F 〈D1, D2, . . . , Dn−1〉 and dimF Ker(adDn) = n− 1.
The Jordan matrix of the nilpotent operator adDn over F has n−
1 Jordan blocks. Denote by J1, J2, . . . , Jn−1 the corresponding Jordan
chains. Without loss of generality, we may take D1 ∈ J1, D2 ∈ J2, . . . ,
Dn−1 ∈ Jn−1 to be the first elements in the corresponding Jordan bases.
If dimF F 〈J1〉 = dimF F 〈J2〉 = · · · = dimF F 〈Jn−1〉 = 1, then FL =
F 〈D1, D2, . . . , Dn〉 and FL is an abelian Lie algebra. It is the algebra
from the conditions of the theorem if s = 0.
Let
dimF F 〈J1〉 > dimF F 〈J2〉 > · · · > dimF F 〈Jn−1〉
and dimF F 〈J1〉 = s + 1, s > 1. Write the elements of the basis J1 as
follows:
J1 =
{
D1,
n−1∑
i=1
a1iDi,
n−1∑
i=1
a2iDi, . . . ,
n−1∑
i=1
asiDi
}
.
By the definition of a Jordan basis,
D1 = [Dn,
n−1∑
i=1
a1iDi] =
n−1∑
i=1
Dn(a1i)Di
K. Sysak 159
whence Dn(a11) = 1 and Dn(a1i) = 0 for all i 6= 1.
By Remark 1,
n−1∑
i=1
a1iDi ∈ FI implies a1i ∈
n−1⋂
k=1
KerDk, i = 1, . . . , n−
1, and thus a12, a13, . . . , a1,n−1 ∈ F, and a11 6∈ F . Let us write a11 = a.
Then a11, a12, . . . , a1,n−1 ∈ F 〈1, a〉.
We shall show that a21, a22, . . . , a2,n−1 ∈ F 〈1, a, a2
2! 〉. By the definition
of a Jordan basis,
[Dn,
n−1∑
i=1
a2iDi] =
n−1∑
i=1
Dn(a2i)Di =
n−1∑
i=1
a1iDi
whence Dn(a2i) = a1i ∈ F 〈1, a〉 for i = 1, . . . , n− 1. Then, by Lemma 7,
a2i ∈
n−1⋂
k=1
KerDk implies a2i ∈ F 〈1, a, a2
2! 〉, i = 1, . . . , n− 1. Assume that
ami ∈ F 〈1, a, . . . , am
m! 〉 for all m = 1, . . . , s− 1 and i = 1, . . . , n− 1. Then
[Dn,
n−1∑
i=1
am+1,iDi] =
n−1∑
i=1
amiDi
whence Dn(am+1,i) = ami for i = 1, . . . , n − 1. The coefficients am+1,i
satisfy the conditions of Lemma 7, so that am+1,i ∈ F 〈1, a, . . . , am+1
(m+1)!〉.
Reasoning by induction, we get that all coefficients aji, i = 1, . . . , n− 1,
j = 1, . . . , s, of the elements from the basis J1 belong to F 〈1, a, . . . , as
s! 〉,
and thus F 〈J1〉 ⊆ L̃.
Consider the basis
J2 =
{
D2,
n−1∑
i=1
b1iDi,
n−1∑
i=1
b2iDi, . . . ,
n−1∑
i=1
btiDi
}
,
where 1 6 t + 1 6 s and dimF F 〈J2〉 = t + 1. By the definition of a
Jordan basis, [Dn,
n−1∑
i=1
b1iDi] =
n−1∑
i=1
Dn(b1i)Di = D2, and thus Dn(b12) =
1 and Dn(b1i) = 0 for all i 6= 2. Set b12 = b 6∈ F and considerDn(b−a) = 0.
It follows from Remark 1 that a, b ∈
n−1⋂
i=1
KerDi, so b− a = δ ∈ F . The
latter means that b ∈ F 〈1, a〉. Moreover, b1i ∈ F for i 6= 2 in view of
Remark 1. Thus, b11, b12, . . . , b1,n−1 ∈ F 〈1, a〉. Reasoning as for J1 and
using Lemma 7, one can show that b2i ∈ F 〈1, a, a2
2! 〉 and prove by induction
that bji ∈ F 〈1, a, . . . , at
t! 〉 for all j = 1, . . . , t and i = 1, . . . , n − 1. Since
t 6 s, we have F 〈J2〉 ⊆ L̃.
160 On nilpotent Lie algebras of derivations
In the same way, one can show that the subspaces F 〈J3〉, F 〈J4〉, . . . ,
F 〈Jn−1〉 lie in L̃. Therefore, the Lie algebra FL is contained in the Lie
subalgebra L̃ of W (A).
Theorem 2. Let L be a nilpotent Lie subalgebra of the Lie algebra W (A),
and let F = F (L) be its field of constants. If L is of rank n > 3 and its
center Z(L) is of rank n− 1 over R, then the Lie algebra FL over F is
isomorphic to a finite dimensional subalgebra of the Lie algebra un(F ) of
triangular polynomial derivations.
Proof. By Theorem 1, the Lie algebra FL is contained in the Lie subal-
gebra L̃ of W (A), which is of the form F 〈D1, aD1,
a2
2!D1, . . . ,
as
s!D1, D2,
aD2, . . . ,
as
s!D2, . . . , Dn−1, . . . ,
as
s!Dn−1, Dn〉, where [Di, Dj ] = 0 for i, j =
1, . . . , n, Dn(a) = 1 and D1(a) = D2(a) = · · · = Dn−1(a) = 0. The Lie
algebra L̃ is isomorphic (as a Lie algebra over F ) to the subalgebra
F
〈
∂
∂x1
, xn
∂
∂x1
,
x2
n
2!
∂
∂x1
, . . . ,
xs
n
s!
∂
∂x1
, . . . ,
∂
∂xn−1
, xn
∂
∂xn−1
, . . . ,
xs
n
s!
∂
∂xn−1
,
∂
∂xn
〉
of the Lie algebra un(F ) of triangular polynomial derivations over F .
2. Example of a maximal nilpotent Lie subalgebra of the
Lie algebra W̃n(K)
Lemma 8 ([8, Lemma 4]). Let K be an algebraically closed field of
characteristic zero. For a rational function φ ∈ K(t), write φ′ = dφ
dt
. If
φ ∈ K(t) \ K, then does not exist a function ψ ∈ K(t) such that ψ′ = φ′
φ
.
Let us denote by K[X] = K[x1, x2, . . . , xn] the polynomial algebra, by
K(X) = K(x1, x2, . . . , xn) the field of rational functions in n variables
over K, and by W̃n(K) the Lie algebra of derivations on the field K(X).
We think that the first part of the following statement is known.
Proposition 1. The subalgebra L = K〈x1
∂
∂x1
, x2
∂
∂x2
, . . . , xn
∂
∂xn
〉 of the
Lie subalgebra of W̃n(K) is isomorphic to a Lie subalgebra of the Lie
algebra un(K) of triangular polynomial derivations, but is not conjugated
with any Lie subalgebra of this Lie algebra by an automorphism of the Lie
algebra W̃n(K).
K. Sysak 161
Proof. Let us show that L is a maximal nilpotent Lie subalgebra of
W̃n(K). Obviously, L is abelian, and so it is nilpotent. Let us show that
L coincides with its normalizer in W̃n(K), which will imply that L is
maximal nilpotent (in view of the well-known fact from the theory of Lie
algebras that a proper Lie subalgebra of a nilpotent Lie algebra does not
coincide with its normalizer, see [1, p.58]).
Let D be an arbitrary element of the normalizer N = N
W̃n(K)
(L).
Then [D,xi
∂
∂xi
] ∈ L for each i = 1, . . . , n. D can be uniquely written as
D =
n∑
j=1
fj
∂
∂xj
, where f1, . . . , fn ∈ K(X). Using the following equations
[
xi
∂
∂xi
,
n∑
j=1
fj
∂
∂xj
]
=
n∑
j=1
[
xi
∂
∂xi
, fj
∂
∂xj
]
=
=
n∑
j=1
j 6=i
xi
∂fj
∂xi
∂
∂xj
+
(
xi
∂fi
∂xi
− fi
) ∂
∂xi
,
we obtain that
xi
∂fj
∂xi
= αjxj , i 6= j, and xi
∂fi
∂xi
− fi = αixi (1)
for αi, αj ∈ K, i, j = 1, . . . , n. We rewrite the first equation in (1) in the
form
∂fj
∂xi
=
αjxj
xi
and consider fj as a rational function in xi over the field
K(x1, . . . , xi−1, xi+1, . . . , xn). By Lemma 8 with φ = φ(xi) = xi, we have
αj = 0. Thus,
∂fj
∂xi
= 0 for all i 6= j. This means that fj ∈ K(xj) for each
j = 1, . . . , n.
Write fi = ui
vi
, where ui, vi ∈ K[xi] are relatively prime and vi 6= 0.
Then the second equation in (1) is rewritten as
xi
u′
ivi − uiv
′
i − αiv
2
i
v2
i
=
ui
vi
,
where ′ denotes the derivative with respect to the variable xi. But then
xi(u
′
ivi − uiv
′
i − αiv
2
i )vi = uiv
2
i , whence we have that the polynomial vi
must divide v′
i. It is possible only if vi ∈ K
∗, i.e. fi is a polynomial in
xi with coefficients in K. Since xi(f
′
i − αi) = fi, we have that fi is a
polynomial of degree 1. It is easy to see that fi = γixi with γi ∈ K for all
i = 1, . . . , n. Thus D ∈ L, that is, L = N and L is a maximal nilpotent
Lie subalgebra of W̃n(K).
162 On nilpotent Lie algebras of derivations
If L is conjugated by an automorphism of the Lie algebra W̃n(K) with
some Lie subalgebra of un(K), then L is contained in a nilpotent Lie
subalgebra of un(K). Therefore, L is not coincide with its normalizer in
W̃n(K), which contradicts the fact proved above. However, the subalgebra
K〈 ∂
∂x1
, . . . , ∂
∂xn
〉 of the Lie algebra un(K) is obviously isomorphic to L.
References
[1] Yu. A. Bahturin, Identical Relations in Lie Algebras (in Russian), Nauka, Moscow
(1985).
[2] V. V. Bavula, Lie algebras of triangular polynomial derivations and an isomorphism
criterion for their Lie factor algebras Izv. RAN. Ser. Mat., 77 (2013), Issue 6, 3-44.
[3] V. V. Bavula, The groups of automorphisms of the Lie algebras of triangular
polynomial derivations, J. Pure Appl. Algebra, 218 (2014), Issue 5, 829-851.
[4] J. Draisma, Transitive Lie algebras of vector fields: an overview, Qual. Theory Dyn.
Syst., 11 (2012), no. 1, 39-60.
[5] A. González-López, N. Kamran and P. J. Olver, Lie algebras of differential operators
in two complex variables, Amer. J. Math., 114 (1992), 1163-1185.
[6] S. Lie, Theorie der Transformationsgruppen, Vol. 3, Leipzig (1893).
[7] Ie. O. Makedonskyi and A. P. Petravchuk, On nilpotent and solvable Lie algebras
of derivations, Journal of Algebra, 401 (2014), 245-257.
[8] Ie. O. Makedonskyi and A. P. Petravchuk, On finite dimensional Lie algebras of
planar vector fields with rational coefficients, Methods Func. Analysis Topology, 19
(2013), no. 4, 376-388.
Contact information
K. Sysak Department of Algebra and Mathematical Logic,
Faculty of Mechanics and Mathematics,
Kyiv National Taras Shevchenko University, 64,
Volodymyrska street, 01033 Kyiv, Ukraine
E-Mail(s): sysakkya@gmail.com
Received by the editors: 24.12.2015
and in final form 10.02.2016.
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