Том 2

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Solvable Lie algebras of derviations of rank one
(2019) Petravchuk, Anatolii; Sysak, Kateryna
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.