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
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
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