Lie algebras of derivations with large abelian ideals
We study subalgebras L of rank m over R of the Lie algebra Wn(K) with an abelian ideal I ⊂ L of the same rank m over R.
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irk-123456789-1884812023-03-03T01:27:15Z Lie algebras of derivations with large abelian ideals Klymenko, I.S. Lysenko, S.V. Petravchuk, A. We study subalgebras L of rank m over R of the Lie algebra Wn(K) with an abelian ideal I ⊂ L of the same rank m over R. 2019 Article Lie algebras of derivations with large abelian ideals / I.S. Klymenko, S.V. Lysenko, A. Petravchuk // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 1. — С. 123–129. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC: Primary 17B66; Secondary 17B05, 13N15. http://dspace.nbuv.gov.ua/handle/123456789/188481 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We study subalgebras L of rank m over R of the Lie algebra Wn(K) with an abelian ideal I ⊂ L of the same rank m over R. |
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Klymenko, I.S. Lysenko, S.V. Petravchuk, A. |
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Klymenko, I.S. Lysenko, S.V. Petravchuk, A. Lie algebras of derivations with large abelian ideals Algebra and Discrete Mathematics |
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Klymenko, I.S. Lysenko, S.V. Petravchuk, A. |
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Lie algebras of derivations with large abelian ideals |
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Lie algebras of derivations with large abelian ideals |
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Lie algebras of derivations with large abelian ideals |
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Lie algebras of derivations with large abelian ideals |
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Lie algebras of derivations with large abelian ideals |
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lie algebras of derivations with large abelian ideals |
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Інститут прикладної математики і механіки НАН України |
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Lie algebras of derivations with large abelian ideals / I.S. Klymenko, S.V. Lysenko, A. Petravchuk // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 1. — С. 123–129. — Бібліогр.: 7 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT klymenkois liealgebrasofderivationswithlargeabelianideals AT lysenkosv liealgebrasofderivationswithlargeabelianideals AT petravchuka liealgebrasofderivationswithlargeabelianideals |
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“adm-n3” — 2019/10/20 — 9:35 — page 123 — #125
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 28 (2019). Number 1, pp. 123–129
c© Journal “Algebra and Discrete Mathematics”
Lie algebras of derivations
with large abelian ideals
I. S. Klymenko, S. V. Lysenko, and A. P. Petravchuk
Abstract. Let K be a field of characteristic zero, A =
K[x1, . . . , xn] the polynomial ring and R = K(x1, . . . , xn) the field
of rational functions. The Lie algebra W̃n(K) := DerK R of all K-
derivation on R is a vector space (of dimension n) over R and every
its subalgebra L has rank rkR L = dimR RL. We study subalgebras
L of rank m over R of the Lie algebra W̃n(K) with an abelian
ideal I ⊂ L of the same rank m over R. Let F be the field of
constants of L in R. It is proved that there exist a basis D1, . . . , Dm
of FI over F , elements a1, . . . , ak ∈ R such that Di(aj) = δij ,
i = 1, . . . ,m, j = 1, . . . , k, and every element D ∈ FL is of the form
D =
∑m
i=1
fi(a1, . . . , ak)Di for some fi ∈ F [t1, . . . tk], deg fi 6 1.
As a consequence it is proved that L is isomorphic to a subalgebra
(of a very special type) of the general affine Lie algebra affm(F ).
Introduction
Let K be a field of characteristic zero,A = K[x1, . . . , xn] the polynomial
ring and R = K(x1, . . . , xn) the field of rational functions in n variables.
The Lie algebra W̃n(K) := DerKR of all K-derivation on R is of great
interest because in case K = R, the field of real numbers, elements of
W̃n(K) (which are of the form
D = f1
∂
∂x1
+ · · ·+ fn
∂
∂xn
, fi ∈ R)
2010 MSC: Primary 17B66; Secondary 17B05, 13N15.
Key words and phrases: Lie algebra, vector field, polynomial ring, abelian ideal,
derivation.
“adm-n3” — 2019/10/20 — 9:35 — page 124 — #126
124 Lie algebras of derivations
can be considered as vector fields on the manifold K
n with rational
coefficients f1, . . . , fn ∈ R. Note that in case K = C or K = R the Lie
algebras W̃1(K) and W̃2(K) were studied by S. Lie [7], A. González-López,
N. Kamran and P. J. Olver [2] and others from the viewpoint of structure
of finite-dimensional subalgebras.
Since W̃n(K) is a vector space of dimension n over R one can define the
rank rkR L over R for any subalgebra L ⊆ W̃n(K) by the rule: rkR L :=
dimR RL. We study subalgebras L ⊆ W̃n(K) of rank m over R which have
an abelian ideal I of the same rank m over R. A natural basis over F ,
the field of constants for L in R, for such Lie algebras is built. Note that
analogous results in cases n = 2 and n = 3 were obtained in [3] and
in [1], in case m = n such a basis can be built using results of [6]. As
a corollary one can prove that the Lie algebra FL over the field F can
be isomorphically embedded into the general affine Lie algebra affm(F ).
This result can be used to study solvable finite dimensional subalgebras
L ⊆ W̃n(K) because such Lie algebras (over an algebraically closed field
of characteristic zero) have a series of ideals
0 ⊂ L1 ⊂ L2 ⊂ . . . ⊂ Lm = L with rkR Ls = s, s = 1, . . . ,m.
We use standard notation. The ground field K is arbitrary of characteristic
zero. Recall that the general affine Lie algebra affm(K) is the semidirect
product affm(K) = glm(K) ⋌ Vm, where Vm is a vector space over K of
dimension m with a zero multiplication and the general linear Lie algebra
glm(K) acts on Vm in the natural way. If L ⊆ W̃n(K) is a subalgebra, then
the field of constants for L in R is the subfield of the field R of the form
F (L) = {r ∈ R | D(r) = 0 for all D ∈ L}.
1. Preliminary results
The next two lemmas contain some technical results about derivations
(see for example, [5] or [3]).
Lemma 1. Let D1, D2 ∈ W̃n(K) and a, b ∈ R. Then:
1) [aD1, bD2] = ab[D1, D2] + aD1(b)D2 − bD2(a)D1,
2) if [D1, D2] = 0, then [aD1, bD2] = aD1(b)D2 − bD2(a)D1,
3) if a, b ∈ KerD1nKerD2, then [aD1, bD2] = ab[D1, D2].
Let L be a subalgebra of W̃n(K) and F = F (L) its field of constants.
Then the set FL of all linear combinations of elements aD, where a ∈ F ,
D ∈ L is a Lie algebra over the field F .
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I . S. Klymenko, S. V. Lysenko, A. P. Petravchuk 125
Lemma 2. If L is an abelian, nilpotent or solvable subalgebra of W̃n(K),
then so is FL respectively.
Lemma 3. Let L be a subalgebra of rank m > 1 over R of the Lie algebra
W̃n(K) and let L contain a proper abelian ideal I of the same rank m over
R. If an inner derivation adT for some T ∈ L is of rank k on the F -
space FI (as a linear operator), then there exist a basis T1, . . . , Tm of FI
over F and elements a1, . . . , ak ∈ R such that Ti(aj) = δij, i = 1, . . . ,m,
j = 1, . . . , k. Besides, T can be written in the form
T = f1(a1, . . . , ak)T1 + . . .+ fm(a1, . . . , ak)Tm,
for some fi ∈ F [t1, . . . , tk], deg fi 6 1, i = 1, . . . ,m.
Proof. Choose any basis D1, . . . , Dm of the vector space FI over F . Since
by [3] (Lemma 3) rkR I = dimF FI it holds T = a1D1 + . . . + amDm
for some elements ai ∈ R. Without loss of generality one can assume
that [D1, T ], . . . , [Dk, T ] form a basis of the vector space T (FI) = [T, FI]
(recall that the linear operator adT is of rank k on FI by the conditions of
the lemma). Any element [Ds, T ], k + 1 6 s 6 m, is a linear combination
of [D1, T ], . . . , [Dk, T ] over F , so we can choose Ds in such a way that
[Ds, T ] = 0. The latter means that in this basis the matrix B = (Di(aj))
is of the form
B =
D1(a1) . . . D1(am)
.
.
Dk(a1) . . . Dk(am)
0 . . . 0
. . . . .
0 . . . 0
and the first k rows R1, . . . , Rk of B are linearly independent over the
field F . Since the matrix B is of rank k over F we can choose k columns
Ci1 , . . . , Cik of B which are linearly independent over F . It is easy to see
that there exists a linear combination γ11R1 + · · ·+ γk1Rk of the first k
rows R1, . . . , Rk of the matrix B such that
γ11R1 + · · ·+ γk1Rk = (∗, . . . 1︸︷︷︸
i1
, ∗, . . . 0︸︷︷︸
i2
, ∗ . . . , 0︸︷︷︸
ik
, . . . ∗),
where the right side is the row with 1 on i1st place, 0 on the i2nd place,
. . . , 0 on the ikth place. Denote D′
1 = γ11D1 + · · ·+ γk1Dk. Then
[D′
1, T ] = r1D1 + · · ·+ 1 ·Di1 + · · ·+ 0 ·Di2 + · · ·+ 0 ·Dik + · · ·+ rmDm
“adm-n3” — 2019/10/20 — 9:35 — page 126 — #128
126 Lie algebras of derivations
for some ri ∈ R, i /∈ {i1, . . . , ik}. The latter means that
D′
1(ai1) = 1, D′
1(ai2) = 0, . . . , D′
1(aik) = 0.
Analogously one can build D′
2, . . . , D
′
k with properties D′
j(ais) = δjs,
s = 1, . . . , k. So we now have a basis D′
1, . . . , D
′
k, Dk+1, . . . , Dm of the vec-
tor space FI over F . Denote for convenience T1 = D′
1, . . . Tk = D′
k, Tk+1 =
Dk+1, . . . , Tm = Dm. Then we have Tj(ais) = δjs, j, s = 1, . . . , k. Be-
sides, by the choice of the initial basis of the vector space FI it holds
Tk+1(ais) = 0, . . . , Tm(ais) = 0, s = 1, . . . , k.
Further any column Cj , j /∈ {i1, . . . , is} is a linear combination of
the columns Ci1 , . . . , Cik of the matrix B, so we can write down Cj =
β1jCi1 + · · ·+ βkjCik for some βij ∈ F . Then
Dt(aj −
k∑
s=1
βsjais) = 0, t = 1, . . . ,m
and therefore aj =
∑k
s=1
βsjais + δj , δj ∈ F . The latter means, that all
the coefficients aj are of the form
aj = fj(ai1 , . . . , aik), fj ∈ F [t1, . . . , tk], deg fj 6 1.
After renumbering the elements ai1 , . . . , aik we get the proof of the last
part of the lemma. The proof is complete.
Lemma 4. Let L be a subalgebra of rank m over R of the Lie algebra
W̃n(K) with an abelian ideal I of the same rank m over R and D ∈ FL. If
there exist a basis D1, . . . , Dm of FI over F and elements a1, . . . , ak ∈ R
with Di(aj) = δij, i = 1, . . . ,m, j = 1, . . . , k, then there exists an element
D ∈ W̃n(K) such that [D −D,Di] = 0, i = 1, . . . , k.
Proof. Since D1, . . . , Dm is a basis of L over R (see Lemma 3 in [3]) the
element D can be written in the form
D = s1D1 + · · ·+ smDm for some si ∈ R.
Then
[Di, D] = Di(s1)D1 + · · ·+Di(sm)Dm
and therefore Di(sj) ∈ F because [Di, D] ∈ FI.
Denote αij = Di(sj), i, j = 1, . . . ,m and consider elements fj =∑k
s=1
αijai, j = 1, . . . ,m. Then Di(fj) = αij , i, j = 1, . . . ,m and therefore
Di(sj − fj) = 0, i = 1, . . . , k, j = 1, . . . ,m. The latter means that
[Di, D −D] = 0, i = 1, . . . , k.
“adm-n3” — 2019/10/20 — 9:35 — page 127 — #129
I . S. Klymenko, S. V. Lysenko, A. P. Petravchuk 127
2. The main result
Theorem 1. Let L be a subalgebra of rank m over R of the Lie algebra
W̃n(K) with a proper abelian ideal I ⊂ L of the same rank m over R and
F be the field of constants for L. Then there exist a basis D1, . . . , Dm
of the ideal FI over F and elements a1, . . . , ak ∈ R, k > 1 such that
Di(aj) = δij, i = 1, . . . ,m, j = 1, . . . , k. Every element D ∈ FL can be
written in the form D =
∑m
i=1
fi(a1, . . . , ak)Di for some linear polynomials
fi ∈ F [t1, . . . , tk].
Proof. Take any element D ∈ L\I. Then the inner derivation adD on FL
is nonzero and by Lemma 3 there exist a basis D1, . . . , Dm of the vector
space FI over F and elements a1, . . . , ak1 ∈ R such that Di(aj) = δij ,
i = 1, . . . ,m, j = 1, . . . , k1 (here k1 is the rank of the linear operator adD
on FI). By the same Lemma 3 the element D can be written in the form
D = f1(a1, . . . , ak1)D1 + · · ·+ fm(a1, . . . , ak1)Dm (1)
for some linear polynomials fi ∈ F [t1, . . . , tk1 ]. If every element of the Lie
algebra FL can be expressed in such a form, then we put k = k1 and the
proof is complete. Let T ∈ FL be any element that is not of form (1).
Then by Lemma 4 there exists an element
T =
m∑
i=1
giDi,
where gi = gi(a1, . . . , ak1), gi ∈ F [t1, . . . , tk1 ] and deg gi 6 1, such that
[Di, T − T ] = 0, i = 1, . . . , k1. (2)
Without loss of generality one can assume that [Di, T ] = 0, i = 1, . . . , k1.
The element T can be written in the form
T = s1D1 + · · ·+ smDm, si ∈ R, i = 1, . . . ,m.
Then the matrix B = (Di(sj)) is of the form
B =
0 . . . 0
. . . . .
0 . . . 0
Dk1+1(s1) . . . Dk1+1(sm)
.
.
Dm(s1) . . . Dm(sm)
“adm-n3” — 2019/10/20 — 9:35 — page 128 — #130
128 Lie algebras of derivations
because [Di, T ] =
∑m
j=1
Di(sj)Dj = 0, i = 1, . . . , k1, and therefore
Di(sj) = 0, i = 1, . . . , k1, j = 1, . . . ,m.
The matrix B is nonzero because the derivation adT is a nonzero
linear operator on the vector F -space FI. Using Lemma 3 one can find
D′
k1+1
, . . . , D′
m ∈ FI and ak1+1, . . . , ak1+k2 ∈ R such that
D′
i(aj) = δij , j = k1 + 1, . . . , k1 + k2, i = 1, . . . ,m.
One can easy to see that D′
k1+1
, . . . , D′
k1+k2
are linear combinations of
the derivations Dk1 , . . . , Dm. Returning to the old notation we can write
Dk1+1 = D′
k1+1
, . . . , Dm = D′
m. Then Di(aj) = δij , i = 1, . . . ,m, j =
1, . . . , k1 + k2. By Lemma 3 the element T can be written in the form
T =
m∑
i=1
fi(a1, . . . , ak1+k2)Di, (3)
where fi ∈ F [t1, . . . , tk1+k2 ], deg fi 6 1. If every element of FL is of
the form (3), then all is done. If not, then we can repeat the above
considerations and build elements
Dk1+k2+1, . . . , Dk1+k2+k3 ∈ FL, ak1+k2+1, . . . , ak1+k2+k3 ∈ R
with properties Di(aj) = δij , i = 1, . . . ,m, j = 1, . . . , k1 + k2 + k3. This
process eventually stops and we get the needed basis D1, . . . , Dm of the
ideal FL, some elements a1, . . . , ak ∈ R with the property Di(aj) = δij
and possibility to write any element of FL in the form
D =
m∑
i=1
fi(a1, . . . , ak)Di,
where fi ∈ F [t1, . . . , tk], deg fi 6 1. The proof is complete.
Corollary 1. Let L be a subalgebra of rank m over R of the Lie algebra
W̃n(K). If L contains an abelian ideal I of the same rank m over R, then
FL is isomorphic to a subalgebra of the general affine Lie algebra affm(F ).
Proof. By Theorem 1 every element D ∈ FL can be written in the form
D = f1(a1, . . . , ak)D1 + · · ·+ fm(a1, . . . , ak)Dm, fi ∈ F [t1, . . . , tk],
with deg fi 6 1, Di(aj) = δij , i = 1, . . . ,m, j = 1, . . . , k. The linear
polynomial fi can be written in the form fi = f i + ci, where ci ∈ F , f i is
“adm-n3” — 2019/10/20 — 9:35 — page 129 — #131
I . S. Klymenko, S. V. Lysenko, A. P. Petravchuk 129
a homogeneous polynomial of degree 1, i.e. a linear form f i =
∑k
j=1
aijxj .
One can establish a correspondence ϕ between the Lie algebra FL and a
subalgebra of the Lie algebra glm(F ) by the rule: if D ∈ FL is of the form
D =
m∑
i=1
fiDi =
m∑
i=1
(
k∑
j=1
aijxj + ci)Di,
then ϕ(D) = A+ c, where
A = (aij) ∈ glm(F ) and c = (c1, . . . , cm) ∈ Vm.
One can easily verify that this correspondence is an injective homomor-
phism from the Lie algebra FL into the general affine Lie algebra affm(F ).
Therefore FL is isomorphic to a subalgebra of the general affine Lie
algebra affm(F ).
References
[1] Ie.Chapovskyi, D.Efimov, A.Petravchuk, Solvable Lie algebras of derivations of
polynomial rings in three variables, Applied problems of Mechanics and Mathe-
matics, Lviv, issue. 16. (2018), 7–13.
[2] A. González-López, N. Kamran and P.J. Olver, Lie algebras of differential operators
in two complex variavles, Amer. J. Math., v.114 (1992), 1163–1185.
[3] Ie. O. Makedonskyi and A.P. Petravchuk, On nilpotent and solvable Lie algebras
of derivations, J. Algebra, 401 (2014), 245–257.
[4] I.S Klimenko, S.V. Lysenko, A.P. Petravchuk, Lie algebras of derivations with
abelian ideals of maximal rank, Uzhgorod University, 31, issue 2 (2017), 83–90 (in
Ukrainian).
[5] A. Nowicki, Polynomial Derivations and their Rings of Constants, Uniwersytet
Mikolaja Kopernika, Torun (1994).
[6] A. Nowicki, Commutative basis of derivations in polynomial and power series rings,
J. Pure Appl. Algebra, 40 (1986), 279–283.
[7] S. Lie, Theorie der Transformationsgruppen, Vol. 3, Leipzig, 1893.
Contact information
I. S. Klymenko,
S. V. Lysenko,
A. P. Petravchuk
Taras Shevchenko National University of Kyiv,
64, Volodymyrska street, 01033 Kyiv, Ukraine
E-Mail(s): ihorKlim93@gmail.com,
svetlana.lysenko.1988@gmail.com,
apetrav@gmail.com
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