Solvable Lie Algebras of Derivations of Rank One

Anatolii Petravchuk,Kateryna Sysak

doi:10.18523/2617-7080220196-10

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 in n variables over K. The Lie algebra Wn(K) of all K-derivations on A is of great interest since its elements may be considered as vector fields on Kn with polynomial coefficients. If L is a subalgebra of Wn(K), then one can define the rank rkAL of L over A as the dimension of the vector space RL over the field R. Finite dimensional (over K) subalgebras of Wn(K) of rank 1 over A were studied by the first author jointly with I. Arzhantsev and E. Makedonskiy. We study solvable subalgebras L of Wn(K) with rkAL = 1, without restrictions on dimension over K. Such Lie algebras are described in terms of Darboux polynomials.

Highlights

Let K be a field of characteristic zero and A = K[x1, . . . , xn] the polynomial ring over K
The Lie algebra Wn(K) is of great interest since its elements may be considered as vector fields on Kn with polynomial coefficients
If L is a subalgebra of the Lie algebra Wn(K), one can define the rank rkA L of L over A as the dimension dimR RL of the vector space RL consisting of all linear combinations of elements aD, where a ∈ R, D ∈ L

Summary

SOLVABLE LIE ALGEBRAS OF DERIVATIONS OF RANK ONE

Xn] the polynomial ring and R = K(x1, . The Lie algebra Wn(K) of all K-derivations on A is of great interest since its elements may be considered as vector fields on Kn with polynomial coefficients. If L is a subalgebra of Wn(K), one can define the rank rkAL of L over A as the dimension of the vector space RL over the field R. Finite dimensional (over K) subalgebras of Wn(K) of rank 1 over A were studied by the first author jointly with I. We study solvable subalgebras L of Wn(K) with rkAL = 1, without restrictions on dimension over K. Such Lie algebras are described in terms of Darboux polynomials

Introduction

The set of partial derivations

The set

Abelian Lie algebras of rank one

From the last relations we obtain

Then h