Solvable Lie algebras of derivations of rank one

Let K be a field of characteristic zero and A = K[x1, . . . , xn] the polynomial ring over K. A KderivationD of A is a K-linear mappingD : A→ A that satisfies the rule: D(ab) = D(a)b+aD(b) for all a, b ∈ A. The set Wn(K) of all K-derivations of the polynomial ring A forms a Lie algebra over K. This Lie algebra is simultaneously a free module over A with the standard basis { ∂ ∂x1 , ∂ ∂x2 , . . . , ∂ ∂xn }. Therefore, for each subalgebra L of Wn(K) one can define the rank rank AL of L over the ring A. Note that for any f ∈ A and D ∈ Wn(K) a derivation fD is defined by the rule: fD(a) = f ·D(a) for all a ∈ A. Finite dimensional subalgebras L of Wn(K) such that rank AL = 1 were described in [1]. We study solvable subalgebras L ⊆ Wn(K) of rank 1 over A without restrictions on the dimension over the field K.


Anatoliy Petravchuk, Kateryna Sysak
Let K be a field of characteristic zero and = K[ 1 , . . . , ] the polynomial ring over K. A Kderivation of is a K-linear mapping : → that satisfies the rule: ( ) = ( ) + ( ) for all , ∈ . The set (K) of all K-derivations of the polynomial ring forms a Lie algebra over K. This Lie algebra is simultaneously a free module over with the standard basis { 1 , 2 , . . . , }. Therefore, for each subalgebra of (K) one can define the rank rank of over the ring . Note that for any ∈ and ∈ (K) a derivation is defined by the rule: ( ) = · ( ) for all ∈ . Finite dimensional subalgebras of (K) such that rank = 1 were described in [1]. We study solvable subalgebras ⊆ (K) of rank 1 over without restrictions on the dimension over the field K.
Recall that a polynomial ∈ is said to be a Darboux polynomial for a derivation ∈ (K) if ̸ = 0 and ( ) = for some polynomial ∈ . The polynomial is called the polynomial eigenvalue of for the derivation . Some properties of Darboux polynomials and their applications in the theory of differential equations can be found in [3]. Denote by the set of all Darboux polynomials for ∈ (K) with the same polynomial eigenvalue and of the zero polynomial. Obviously, the set is a vector space over K. If is a subspace of for any derivation ∈ (K), then we denote by the set of all derivations , ∈ .
Theorem 1. Let be a subalgebra of the Lie algebra (K) of rank 1 over and dim K ≥ 2. The Lie algebra is abelian if and only if there exist a derivation ∈ (K) and a Darboux polynomial for with the polynomial eigenvalue such that = for some K-subspace ⊆ .
Using this result one can characterize nonabelian subalgebras of rank 1 over of the Lie algebra (K). For the Lie algebrã︁ (K) of all K-derivations of the field K( 1 , 2 , . . . , ) this problem is simpler and was considered in [2].
The set of all ternary invertible functions is denoted by ∆ 3 . If an operation is invertible and (14) , (24) , (34) are its inverses in those semigroups, then the algebra ( ; , (14) , (24) , (34) ) (in brief, ( ; )) is called a ternary quasigroup [1]. The inverses are also invertible. All inverses to inverses are called -parastrophes of the operation and can be defined by where 4 denotes the group of all bijections of the set {0, 1, 2, 3}. Therefore in general, every invertible operation has 24 parastrophes. Since parastrophes of a quasigroup satisfy the equalities ( ) = , then the symmetric group 4 defines an action on the set ∆ 3 . In particular, the fact implies that the number of different parastrophes of an invertible operation is a factor of 24. More precisely, it is equal to 24/|Ps( )|, where Ps( ) denotes a stabilizer group of under this action which is called parastrophic symmetry group of the operation .
Let P( ) denote the class of all quasigroups whose parastrophic symmetry group contains the group ∈ 4 . A ternary quasigroup ( ; ) belongs to P( ) if and only if = for all from a set of generators of the group , therefore, the class of quasigroup P( ) is a variety.
For every subgroup of the group 4 the variety P( ) are described and its subvariety of ternary group isotopes are found. For example, let