Finite-dimensional subalgebras in polynomial Lie algebras of rank one
Let $W_n(\mathbb{K})$ be the Lie algebra of derivations of the polynomial algebra $\mathbb{K}[X] := \mathbb{K}[x_1,... ,x_n]$ over an algebraically closed field $K$ of characteristic zero. A subalgebra $L \subseteq W_n(\mathbb{K})$ is called polynomial if it is a submodule of the $\mathbb{K}[X]$-mo...
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Institute of Mathematics, NAS of Ukraine
2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508719599583232 |
|---|---|
| author | Arzhantsev, I. V. Makedonskii, E. A. Petravchuk, A. P. Аржанцев, І.В. Македонський, Є. А. Петравчук, А. П. |
| author_facet | Arzhantsev, I. V. Makedonskii, E. A. Petravchuk, A. P. Аржанцев, І.В. Македонський, Є. А. Петравчук, А. П. |
| author_sort | Arzhantsev, I. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:35:13Z |
| description | Let $W_n(\mathbb{K})$ be the Lie algebra of derivations of the polynomial algebra $\mathbb{K}[X] := \mathbb{K}[x_1,... ,x_n]$ over an algebraically closed field $K$ of characteristic zero.
A subalgebra $L \subseteq W_n(\mathbb{K})$ is called polynomial if it is a submodule of the $\mathbb{K}[X]$-module $W_n(\mathbb{K})$.
We prove that the centralizer of every nonzero element in $L$ is abelian provided that $L$ is of rank one.
This fact allows to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one. |
| first_indexed | 2026-03-24T02:29:41Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.554
I. V. Arzhantsev (Moscow State Univ., Russia),
E. A. Makedonskii, A. P. Petravchuk (Kyiv Nat. Taras Shevchenko Univ., Ukraine)
FINITE-DIMENSIONAL SUBALGEBRAS
IN POLYNOMIAL LIE ALGEBRAS OF RANK ONE*
СКIНЧЕННОВИМIРНI ПIДАЛГЕБРИ
ПОЛIНОМIАЛЬНИХ АЛГЕБР ЛI РАНГУ ОДИН
Let Wn(K) be the Lie algebra of derivations of the polynomial algebra K[X] := K[x1, . . . , xn] over an
algebraically closed field K of characteristic zero. A subalgebra L ⊆ Wn(K) is called polynomial if it is
a submodule of the K[X]-module Wn(K). We prove that the centralizer of every nonzero element in L is
abelian provided that L is of rank one. This fact allows to classify finite-dimensional subalgebras in polynomial
Lie algebras of rank one.
Нехай Wn(K) — алгебра Лi диференцiювань полiномiальної алгебри K[X] := K[x1, . . . , xn] над алгеб-
раїчно замкненим полем K характеристики нуль. Пiдалгебра L ⊆ Wn(K) називається полiномiальною,
якщо вона є пiдмодулем K[X]-модуля Wn(K). Доведено, що централiзатор кожного ненульового еле-
мента з L є абелевим у випадку, коли L має ранг 1. Це дає можливiсть класифiкувати скiнченновимiрнi
пiдалгебри полiномiальних алгебр Лi рангу 1.
Introduction. Let K be an algebraically closed field of characteristic zero and K[X] :=
K[x1, . . . , xn] the polynomial algebra over K. Recall that a derivation of K[X] is a
linear operator D : K[X]→ K[X] such that
D(fg) = D(f)g + fD(g) for all f, g ∈ K[X].
Every derivation of the algebra K[X] has the form
P1
∂
∂x1
+ . . .+ Pn
∂
∂xn
for some P1, . . . , Pn ∈ K[X].
A derivation D may be extended to the derivation D of the field of rational functions
K(X) := K(x1, . . . , xn) by
D
(
f
g
)
:=
D(f)g − fD(g)
g2
.
The kernel S of D is an algebraically closed subfield of K(X), cf. [6] (Lemma 2.1).
Denote by Wn(K) the Lie algebra of all derivations of K[X] with respect to the
standard commutator. The study of the structure of the Lie algebra Wn(K) and of
its subalgebras is an important problem appearing in various contexts (note that in
case K = R or K = C we have the Lie algebra Wn(K) of all vector fields with
polynomial coefficients on Rn or Cn). Since Wn(K) is a free K[X]-module
(
with the
basis
∂
∂x1
, . . . ,
∂
∂xn
)
, it is natural to consider the subalgebras L ⊆ Wn(K) which are
*The first author was supported by the RFBR, grant 09-01-90416-Ukr-f-a. The third author was supported
by the DFFD, grant F28.1/026.
c© I. V. ARZHANTSEV, E. A. MAKEDONSKII, A. P. PETRAVCHUK, 2011
708 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
FINITE-DIMENSIONAL SUBALGEBRAS . . . 709
K[X]-submodules. Following the work of V. M. Buchstaber and D. V. Leykin [1], we
call such subalgebras the polynomial Lie algebras. In [1], the polynomial Lie algebras
of maximal rank were considered. Earlier, D. A. Jordan studied subalgebras of the Lie
algebra Der(R) for a commutative ring R which are R-submodules in the R-module
Der(R) (see [4]).
In this note, we study polynomial Lie algebras L of rank one. In Section 2 we prove
that the centralizer of every nonzero element in L is abelian. Clearly, this property is
inherited by any subalgebra in L. It is not difficult to describe all finite-dimensional Lie
algebras with this property, see Proposition 2. In Theorem 1 we give a classification
of finite-dimensional subalgebras in polynomial Lie algebras of rank one: every such
subalgebra is either abelian, or solvable with an abelian ideal of codimension one and
trivial center, or isomorphic to sl2(K). Moreover, for all these three types we construct
an explicit realization in some L. Applying obtained results to the Lie algebra W1(K) we
give a description of all finite dimensional subalgebras of W1(K) (Proposition 3). In case
K = C this description can be easily deduced from classical results of S. Lie (see [5])
about realizations (up to local diffeomorphisms) of finite dimensional Lie algebras by
vector fields on the complex line. In [5], S. Lie has also classified analogous realizations
on the complex plane and on the real line. On the real plane such a classification is given
in [2].
1. Lie algebras with abelian centralizers. We begin with an elementary lemma
on submodules of a free module. Let A be a unique factorization domain and N =
= Ae1 ⊕ . . . ⊕ Aen a free A-module. An element x ∈ N is said to be reduced if the
condition x = ax′ with a ∈ A and x′ ∈ N implies that the element a is invertible in A.
Lemma 1. For every submodule M ⊆ N of rank one there exist an ideal I ⊆ A
and a reduced element m0 ∈ N such that M = Im0. The submodule M defines the
element m0 uniquely up to multiplication by an invertible element of A.
Proof. Take a nonzero element m ∈ M, m = a1e1 + . . . + anen. Let a be the
greatest common divisor of a1, . . . , an, and m0 = a01e1+ . . .+a
0
nen, where a0i = ai/a.
Since M has rank one, for every nonzero m′ ∈M there are nonzero c, d ∈ A such that
cm+dm′ = 0. Then acm0+dm
′ = 0. If m′ = a′1e1+ . . .+a
′
nen, then aca0i +da
′
i = 0
for all i = 1, . . . , n. If d does not divide ac, then some prime p ∈ A divides all the
elements a01, . . . , a
0
n. But the elements a01, . . . , a
0
n are coprime, a contradiction. Thus m′
equals bm0 with b = ac/d. This proves that all elements of M have the form bm0 for
some b ∈ A. Clearly, all elements b ∈ A such that bm0 ∈M form an ideal I of A. The
second assertion follows from the fact that a free A-module has no torsion.
Lemma 1 is proved.
We say that a derivation P1
∂
∂x1
+ . . . + Pn
∂
∂xn
is reduced if the polynomials
P1, . . . , Pn are coprime. Setting A = K[X] and N = Wn(K), we get the following
variant of Lemma 1.
Lemma 2. For every submodule M ⊆ Wn(K) of rank one there exist an ideal
I ⊆ K[X] and a reduced derivation D0 ∈Wn(K) such that M = ID0. The submodule
M defines the derivation D0 uniquely up to nonzero scalar.
Now we study the centralizers of elements in a polynomial Lie algebra of rank one.
Proposition 1. Let L be a subalgebra of the Lie algebra Wn(K). Assume that L
is a submodule of rank one in the K[X]-module Wn(K). Then the centralizer of any
nonzero element in L is abelian.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
710 I. V. ARZHANTSEV, E. A. MAKEDONSKII, A. P. PETRAVCHUK
Proof. By Lemma 2, the subalgebra L has the form ID0 for some reduced derivation
D0 ∈ Wn(K). Denote by D0 the extension of D0 to the field K(X), and let S be the
kernel of D0. Take any nonzero element fD0 ∈ L, f ∈ I, and consider its centralizer
C = CL(fD0). For every nonzero element gD0 ∈ C one has
[fD0, gD0] = (fD0(g)− gD0(f))D0 = 0.
This implies D0(f)g − fD0(g) = 0, thus D0(f/g) = 0 and f/g ∈ S. Take another
nonzero element hD0 ∈ C. By the same arguments we get f/h ∈ S. This shows that
g/h ∈ S. The latter condition is equivalent to [gD0, hD0] = 0, so the subalgebra C is
abelian.
Proposition 1 is proved.
The next proposition seems to be known, but having no precise reference we supply
it with a complete proof. By Z(F ) we denote the center of a Lie algebra F.
Proposition 2. Let F be a finite-dimensional Lie algebra over an algebraically
closed field K of characteristic zero. Assume that the centralizers of all nonzero elements
in F are abelian. Then either F is abelian, or F ∼= Ah 〈b〉, where b ∈ F, A ⊂ F is an
abelian ideal and Z(F ) = 0, or F ∼= sl2(K).
Proof. If the centralizers of all nonzero elements of a Lie algebra F are abelian,
then the same property holds for every subalgebra of F. Assume that F is not abelian
and the centralizers of all elements of F are abelian. Then the center Z(F ) is trivial.
Case 1. F is solvable. Then F contains a non-central one-dimensional ideal 〈a〉,
see [3] (II.4.1, Corollary B). Let A be the centralizer of a in F. Clearly, A is an abelian
ideal of codimension one in F. Then F ∼= Ah 〈b〉 for any b ∈ F \A.
Case 2. F is semisimple. Then F = F1 ⊕ . . . ⊕ Fk is the sum of simple ideals.
Since the centralizer of every element x ∈ F1 contains F2 ⊕ . . .⊕ Fk, we conclude that
F is simple. Let H be a Cartan subalgebra in F and F = N− ⊕ H ⊕ N+ the Cartan
decomposition with opposite maximal nilpotent subalgebras N− and N+ in F, see [3]
(II.8.1). Since the centrilizer of every element in N+ is abelian, either the subalgebra N+
is abelian or Z(N+) = 0. The second possibility is excluded because N+ is nilpotent.
Thus N+ is abelian. This is the case if and only if the root system of the Lie algebra F
has rank one, or, equivalently, F ∼= sl2(K).
Case 3. F is neither solvable nor semisimple. Consider the Levi decomposition
F = R h G, where G is a maximal semisimple subalgebra and R is the radical of F.
By Case 2, the algebra G is isomorphic to sl2(K). Denote by A the ideal of R which
coincides with R if R is abelian, and A = [R,R] otherwise. By Case 1, the ideal A is
abelian. Consider the decomposition A = A1 ⊕ . . . ⊕ As into simple G-modules with
respect to the adjoint representation. If dimA1 = 1, then the centralizer of a nonzero
element in A1 contains G, a contradiction. Suppose that dimA1 ≥ 2. Fix an sl2-triple
{e, h, f} in G and take a highest vector x ∈ A1 with respect to the Borel subalgebra
〈e, h〉. Then [e, x] = 0 and the centralizer CF (x) contains the subalgebra Ah 〈e〉. The
latter is not abelian because the adjoint action of the element e on A1 is not trivial. This
contradiction concludes the proof.
2. Main results. In this section we get a classification of finite-dimensional subalgebras
in polynomial Lie algebras of rank one.
Theorem 1. Let L be a polynomial Lie algebra of rank one in Wn(K), where K
is an algebraically closed field of characteristic zero, and F ⊂ L a finite-dimensional
subalgebra. Then one of the following conditions holds:
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
FINITE-DIMENSIONAL SUBALGEBRAS . . . 711
(1) F is abelian;
(2) F ∼= Ah 〈b〉, where A ⊂ F is an abelian ideal and [b, a] = a for every a ∈ A;
(3) F is a three-dimensional simple Lie algebra, i.e., F ∼= sl2(K).
Proof. By Propositions 1 and 2, every finite-dimensional subalgebra F ⊂ L is either
abelian, or has the form Ah 〈b〉, or is isomorphic to sl2(K). It remains to prove that in
the second case we may find b ∈ F with [b, a] = a for every a ∈ A. Take any element
b with F = Ah 〈b〉.
Let us prove that the operator ad(b) is diagonalizable. Assuming the converse, let
a0, a1 ∈ A be nonzero elements with [b, a1] = λa1 + a0, [b, a0] = λa0 for some λ ∈ K.
By Lemma 2, the subalgebra L has the form ID0 for some ideal I ⊆ K[X] and some
reduced derivation D0 ∈Wn(K). Set a0 = fD0, a1 = gD0, b = hD0, f, g, h ∈ I. The
relations [b, a1] = λa1 + a0, [b, a0] = λa0, and [a0, a1] = 0 are equivalent to
hD0(g)− gD0(h) = λg + f, hD0(f)− fD0(h) = λf, fD0(g)− gD0(f) = 0.
Multiplying the second relation by g, we get hgD0(f)− fgD0(h) = λfg. This and the
third relation imply hfD0(g)− fgD0(h) = λfg ⇒ hD0(g)− gD0(h) = λg. Together
with the first relation it gives f = 0, a contradiction.
Now assume that [b, a1] = λ1a1 and [b, a2] = λ2a2 for some λ1, λ2 ∈ K. If
a1 = fD0, a2 = gD0, b = hD0, then we obtain the relations
hD0(f)− fD0(h) = λ1f, hD0(g)− gD0(h) = λ2g, fD0(g)− gD0(f) = 0.
Consequently,
ghD0(f) = gf(λ1 +D0(h)) = fhD0(g) = fg(λ2 +D0(h)).
This proves that λ1 = λ2 and hence ad(b) is a scalar operator. Since F is not abelian,
ad(b) is nonzero and, multiplying b by a suitable scalar, we may assume that ad(b) is
the identical operator.
Theorem 1 is proved.
Let us show that all three possibilities indicated in Theorem 1 are realizable. Take
a derivation D0 ∈ Wn(K) such that there exist non-constant polynomials p, q ∈ K[X]
with D0(p) = 0 and D0(q) = 1. For example, one may take D0 =
∂
∂x2
+P3
∂
∂x3
+ . . .
. . .+ Pn
∂
∂xn
with arbitrary P3, . . . , Pn ∈ K[X], and p = x1, q = x2.
The subalgebra 〈D0, pD0, . . . , p
m−1D0〉 is an m-dimensional abelian subalgebra in
K[X]D0 for every positive integer m.
The subalgebra A h 〈b〉 with dimA = m may be obtained by setting A =
= 〈D0, pD0, . . . , p
m−1D0〉 and b = −qD0. Indeed,
[−qD0, f(p)D0] = (−D0(f(p))+f(p)D0(q))D0 = f(p)D0 for every f(p) ∈ K[p].
Finally, the derivations e = q2D0, h = 2qD0 and f = −D0 form an sl2-triple in
K[X]D0.
Remark 1. The structure of finite-dimensional subalgebras in a polynomial Lie
algebra L = ID0 depends on properties of the derivation D0. In particular, if
Ker(D0) = K, then all abelian subalgebras in K[X]D0 are one-dimensional.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
712 I. V. ARZHANTSEV, E. A. MAKEDONSKII, A. P. PETRAVCHUK
Our last result concerns finite-dimensional subalgebras in the Lie algebra W1(K).
By Lemma 2, every polynomial Lie algebra in W1(K) has the form L = q(x)K[x]
∂
∂x
with some polynomial q(x) ∈ K[x].
Proposition 3. Let L = q(x)K[x]
∂
∂x
be a polynomial algebra.
1. If deg q(x) ≥ 2, then every finite dimensional Lie subalgebra in L is one-
dimensional.
2. If deg q(x) = 1, then every finite dimensional Lie subalgebra in L is either
one-dimensional or coincides with Fk =
〈
q(x)
∂
∂x
, q(x)k
∂
∂x
〉
for some k ≥ 2.
3. If q(x) = const 6= 0 (i.e., L = W1(K)), then every finite dimensional Lie
subalgebra in L is either one-dimensional, or coincides with Fk,β =
〈
(x+ β)
∂
∂x
, (x+
+ β)k
∂
∂x
〉
for some β ∈ K and k = 0, 2, 3, . . . , or is a three-dimensional subalgebra
F (β) =
〈
∂
∂x
, (x+ β)
∂
∂x
, (x+ β)2
∂
∂x
〉
, where β ∈ K.
Proof. Let us describe all two-dimensional subalgebras in W1(K). Every such
subalgebra has the form〈
f(x)
∂
∂x
, g(x)
∂
∂x
〉
with f(x), g(x) ∈ K[x] and fg′ − f ′g = g. (∗)
If deg(f) ≥ 2, then looking at the highest terms of fg′ and f ′g, we get deg(f) = deg(g).
But the polynomials (f + λg, g) satisfy relation (∗) for every λ ∈ K, and thus we may
assume that f is linear. Each root of g is also a root of f, so g is proportional to fk for
some k = 0, 2, 3, . . . . This observation together with Theorem 1 and Remark 1 proves
all the assertions.
Proposition 3 is proved.
If we consider obtained in Proposition 3 realizations up to automorphisms of the
polynomial ring K[x], then in case deg q(x) = 1 for the Lie algebra Fk one can take
q(x) = x, and in case q(x) = const 6= 0 one can take β = 0.
1. Buchstaber V. M., Leykin D. V. Polynomial Lie algebras // Funk. Anal. Pril. – 2002. – 36, № 4. –
S. 18 – 34 (in Russian) (English transl.: Funct. Anal. and Appl. – 2002. – 36, № 4. – P. 267 – 280.).
2. González-López A., Kamran N., Olver P. J. Lie algebras of vector fields in the real plane // Proc. London
Math. Soc. Third Ser. – 1992. – 64, № 2. – P. 339 – 368.
3. Humphreys J. E. Introduction to Lie algebras and representation theory. – New York: Springer, 1972.
4. Jordan D. A. On the ideals of a Lie algebra of derivations // J. London Math. Soc. – 1986. – 33, № 1. –
P. 33 – 39.
5. Lie S. Theorie der Transformationsgruppen. – Leipzig, 1888, 1890, 1893. – Vols 1 – 3.
6. Nowicki A., Nagata M. Rings of constants for k-derivations of k[x1, . . . , xn] // J. Math. Kyoto Univ. –
1988. – 28, № 1. – P. 111 – 118.
Received 18.05.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
|
| id | umjimathkievua-article-2755 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:29:41Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-27552020-03-18T19:35:13Z Finite-dimensional subalgebras in polynomial Lie algebras of rank one Скiнченновимiрнi пiдалгебри полiномiальних алгебр лi рангу один Arzhantsev, I. V. Makedonskii, E. A. Petravchuk, A. P. Аржанцев, І.В. Македонський, Є. А. Петравчук, А. П. Let $W_n(\mathbb{K})$ be the Lie algebra of derivations of the polynomial algebra $\mathbb{K}[X] := \mathbb{K}[x_1,... ,x_n]$ over an algebraically closed field $K$ of characteristic zero. A subalgebra $L \subseteq W_n(\mathbb{K})$ is called polynomial if it is a submodule of the $\mathbb{K}[X]$-module $W_n(\mathbb{K})$. We prove that the centralizer of every nonzero element in $L$ is abelian provided that $L$ is of rank one. This fact allows to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one. Нехай $W_n(\mathbb{K})$ — алгебра Лi диференцiювань полiномiальної алгебри $\mathbb{K}[X] := \mathbb{K}[x_1,... ,x_n]$ над алгебраїчно замкненим полем $K$ характеристики нуль. Пiдалгебра $L \subseteq W_n(\mathbb{K})$ називається полiномiальною, якщо вона є пiдмодулем $\mathbb{K}[X]$-модуля $W_n(\mathbb{K})$. Доведено, що централiзатор кожного ненульового елемента з $L$ є абелевим у випадку, коли $L$ має ранг 1. Це дає можливiсть класифiкувати скiнченновимiрнi пiдалгебри полiномiальних алгебр Лi рангу 1. Institute of Mathematics, NAS of Ukraine 2011-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2755 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 5 (2011); 708-712 Український математичний журнал; Том 63 № 5 (2011); 708-712 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2755/2266 https://umj.imath.kiev.ua/index.php/umj/article/view/2755/2267 Copyright (c) 2011 Arzhantsev I. V.; Makedonskii E. A.; Petravchuk A. P. |
| spellingShingle | Arzhantsev, I. V. Makedonskii, E. A. Petravchuk, A. P. Аржанцев, І.В. Македонський, Є. А. Петравчук, А. П. Finite-dimensional subalgebras in polynomial Lie algebras of rank one |
| title | Finite-dimensional subalgebras in polynomial Lie algebras of rank one |
| title_alt | Скiнченновимiрнi пiдалгебри полiномiальних алгебр лi рангу один |
| title_full | Finite-dimensional subalgebras in polynomial Lie algebras of rank one |
| title_fullStr | Finite-dimensional subalgebras in polynomial Lie algebras of rank one |
| title_full_unstemmed | Finite-dimensional subalgebras in polynomial Lie algebras of rank one |
| title_short | Finite-dimensional subalgebras in polynomial Lie algebras of rank one |
| title_sort | finite-dimensional subalgebras in polynomial lie algebras of rank one |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2755 |
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