Solvable Lie Algebras of Derivations of Rank One

Anatolii Petravchuk, Kateryna Sysak
2019 Mohyla Mathematical Journal  
Let K be a field of characteristic zero, A = K[x 1 , . . . , x n ] the polynomial ring and R = K(x 1 , . . . , x n ) the field of rational functions in n variables over K. The Lie algebra W n (K) of all K-derivations on A is of great interest since its elements may be considered as vector fields on K n with polynomial coefficients. If L is a subalgebra of W n (K), then one can define the rank rk A L of L over A as the dimension of the vector space RL over the field R. Finite dimensional (over
more » ... dimensional (over K) subalgebras of W n (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 W n (K) with rk A L = 1, without restrictions on dimension over K. Such Lie algebras are described in terms of Darboux polynomials.
doi:10.18523/2617-7080220196-10 fatcat:pe6p2qlciffopflf2e57jif2ze